Hi Michelle! Welcome to the community! It's an honor to have another new member with a lot of parrot expertise and experience! And congrats on your new baby!
I grew-up in a parrot-breeding family and bred/hand-raised/hand-fed myself for about 20 years, and have a graduate degree in Animal Health Science, and have been the medical liaison at an Avian and Reptile Rescue for over 8 years now. Hand-feeding a baby parrot is not an easy task, and we're seeing more and more people buying/bringing home baby parrots that are far from being weaned, and a lot of the time it ends very badly...However, at least in your case you have lots of experience and know what you're doing, so you're way ahead of the ball game.
The first comment I would make to you is to be extremely careful with your formula temperature, because at 45-47 degrees C (113 degrees F-116.6 degrees F) you are way too hot and are going to end-up burning his crop...And crop-burns from hand-feeding formula usually come-on gradually, over multiple feedings with the formula being too hot. And if he does sustain a crop burn, the treatment for it is a frustrating, tedious, and scary one, as they typically have to cut away the burnt tissue, resulting in an actual "hole" in their crop, and usually severe infection results as well and requires long-term antibiotic therapy, which then typically results in a horrible secondary fungal-infection throughout the GI Tract. I've seen this happen several times and it's always horribly awful for both the bird and the owner...So you need to bring down your formula temperature so that it is ALWAYS between 40 degrees Celsius (104 degrees F) and an absolute maximum temperature of 43.3 degrees Celsius (110 degrees F)...Any cooler than 40 degrees C and they develop a fungal (yeast) infection that results in slow-crop and eventually crop-stasis, and any hotter than 43.3 degrees C and they will surely develop gradual crop burns...At the high-end you're talking about (47 degrees C, 116.6 degrees F) you could cause an acute crop burn that would be life-threatening. So that's the first thing you need to change; I'm assuming that you're using some type of candy or cooking thermometer with a metal probe to insert into the formula, and that you keep it in the formula the entire time that you're feeding. And little ProTip: Never microwave the already-mixed formula to reheat it if it falls below 40 degrees C while you're feeding him, but rather heat-up more water/pedialyte and add it to the formula to avoid any "hot pockets" in the formula that also cause crop burns...
Your bird is right around 5 weeks old, so 4 formula feedings a day sounds about right. Most Eckies start to "Abundance Wean", or at the very least start to show signs of starting to Abundance-Wean around 8 weeks old, with many of them fully Abundance Weaning themselves between 11-12 weeks, with 14-15 weeks being the high-end...However, it's extremely important, as I'm sure you already know, to allow your bird to fully Abundance-Wean himself, at HIS PACE, and however long that takes him is fine. Some Eckies can take as long as 6 months to fully Abundance Wean themselves, and while this is extremely rare, if it takes that long then it takes that long. It should never be you that takes away a daily feeding or decreases the amount of formula in a feeding...What you're describing as far as the number of daily feedings he's currently taking and the amounts in each feeding are pretty normal for a 5 week-old Eckie.
I'm not familiar with "Paswells" brand hand-feeding formula, but I'm assuming it's similar to most-all hand-feeding formulas...I understand what you mean about him "not being enthusiastic" about his feedings, and yes, they do usually prefer their formula at the hotter end of the spectrum, but again, I would highly advise you to never go above 43.3 degrees C (110 degrees F), I'm actually quite surprised that he hasn't already suffered some degree of burn to his crop if you have actually fed him formula as hot as 116.6 degrees F (47 degrees C), I've seen very severe crop burns around 114 degrees F, so you're definitely at the end of the spectrum that is dangerous...
Can you actually see his crop at this point (I'm assuming he's got all of his down at this point, including over his crop, if not already covered with pin-feathers)? I would try to take a look at his actual crop, both externally and internally if possible, and look for redness, swelling, any discharge, scabbing, etc. I'm only saying this because of the formula temperatures you've been feeding and the fact that you say he's "not enthusiastic about feeding", which i not really unusual, not necessarily, but crop burns will definitely cause a lack of appetite and eventually anorexia, so I'd take a good look just to be sure.
Otherwise, something else that can help a bit is to make the formula a bit thicker, as they tend to prefer it thicker and thicker the older they get, and will often reject thin/runny formula.
Another question that is extremely important is what ambient temperature are you keeping him at? Again, I'm assuming that at 5 weeks he's got all of his down in by now, and if he does have all of his down feathers in at this point but obviously not all of his outer feathers, then his ambient temperature needs to always be right around 80-85 degrees F; before they get all of their down in they need to be kept at an ambient temperature around 95 degrees F...The consequence of this is obviously very much the same of formula that is cooler than 104 degrees F (40 degrees C), and that is a fungal (yeast) infection in the crop and throughout the GI Tract, slow-crop, and crop-stasis. I don't know what type of Brooder you have him in, whether a proper Brooder or a homemade one, but his ambient temperature is just as important as his formula temperature until he is fully-feathered. Whenever a baby bird isn't very enthusiastic about eating or their feeding-response is lacking, the first thing that is usually wrong is that their ambient temperature hasn't been warm enough, and usually if this is the case, as soon as their ambient temperature is bumped-up to the correct range, they typically start eating their formula with vigor. So if you've had him for only a week, meaning he arrived when he was about 4 weeks-old, he should have had all of his down feathers when he arrived, meaning his ambient temperature since you've had him should be between 80-85 degrees F.
I'm assuming that his crop is almost empty at each time you feed him, and completely empties overnight? Have you smelled his breath (open mouth/beak) to make sure there is no "sour" smell (yeast smell)? If not, I'd just do it to be sure, as a yeast infection in the crop is a common cause of them not wanting to eat, and if his ambient temperature hasn't been warm enough then yeast will definitely grow.
Also, have you started to offer him any solid food yet? I've not ever weaned an Ekkie myself, but I'm aware that they can be very difficult to Abundance-Wean compared to other species of parrots. Assuming that all of what I listed above is correct or corrected, such as the formula temperatures, his ambient temperature (big one), etc., then you need to move-on to basic Eclectus Abundance-Weaning practices. First thing is you should be weighing him every single day on a digital kitchen scale and writing it down with the date, time, and weight, so that you can make sure he is gaining and not losing.
Weighing him first thing in the morning after he poops but before you feed him is the best time to weigh him each day. He should continue to gain weight until he fledges, where they tend to lose a bit of weight. So if he's not gaining weight at 5 weeks-old, then you'll know that something is amiss. If he is continually gaining weight, then you'll know that the amount of formula he's eating is adequate. Another important question for you, since you got him at 4 weeks old is at what age was he pulled from his parents? If he was taken too soon this can cause issues with them feeding and/or weaning properly. With most species of parrots they should not be pulled any sooner than 2 weeks old and no older than 3 weeks old...So if he wasn't pulled until the time when you got him, at 4 weeks old, this would explain why he's not really interested in eating his formula with much vigor. It's okay, it shouldn't cause any long-term issues, but it's an explanation for you if that was the case...
The Ekkie people can help you with what solid foods to start weaning him onto, that's not my area of expertise. Once again, the biggest thing is that he is allowed to Abundance-Wean himself, and it should take him until he is between 3-4 months old to fully Abundance-Wean himself.
Good luck!
