First and foremost, never, EVER buy a bird for your bird. Only buy a bird if YOU want another bird for yourself and have the time to treat it as a pet with its own time/play/space etc. I know your intention is to help your mourning bird, but a new bird will not replace what was lost (as they are all unique and the birds know this)...If you are committed to caring for 3 birds equally in totally separate spheres and cages with separate time for interaction etc, and YOU want to do it and have the time, then do it (as long as you can do it safely in terms of disease etc)..because that is what you could be looking at in the worst-case-scenario.
There are quite a few risks-- 1. if you get another bird, your old bird may not accept him/her at all and then you are looking at 3x the commitment required for one bird, as all 3 will have to be interacted with and allowed free-time while the others are caged--this could easily be 6-8 hours a day for all 3 when combined. They could seriously fight.
2. A new budgie could bring in asymptomatic diseases to your other 2 birds, or, you could spread asymptomatic disease from your current birds to the new one. Many diseases can be carried and spread without symptoms, or take years to show symptoms. Given the fact that you are unsure of your last bird's death, it seems risky. Many viruses cause neurological impacts and can survive on surfaces and undetected in seemingly healthy birds for years.
3. The birds could become hormonal (even if you have 2 of the same sex) and you could end up with new behavioral issues, or be the 3rd wheel in a weird love-triangle. If you get 2 of the opposite sex, they could still fight but they could also mate and then you are going to be in way over your head. With females, you run the risk of egg-binding (even if they are on their own--a hormonal female can lay infertile eggs when she feels like it- and that can pose health risks).
4. Three is often a troublesome number for people or birds.
Can it work? Yes. Does it always---I would say 50% of the time, birds don't get along (and there is a sub-sect of that rough percentage that tolerates the other while still requiring constant supervision during interaction). Even if they seem to get along, housing them together could lead to all sorts of risks....it could work, but it's so complicated.
Even in the same house, birds can spread disease, so quarantine and a better vet than your last would be ESSENTIAL ...I do believe they sell swab kits for certain viruses, but that could be expensive. I would certainly run blood work and panels on your current birds before bringing a new bird into the mix, and then do the same for him while in quarantine.
I WOULD NOT NOT NOT combine a female and male parakeet -PERIOD. It's always going to lead to the risk of hormones and breeding...