Multi-parrot families are COMPLETELY POSSIBLE - I have two conures (probably the easiest to integrate into a flock) and I know a gentleman who has a RFM, a Red-Lored 'Zon, a CAG and three conures and they are one big happy family ... and Riley's Mom has her flock so it's is POSSIBLE without issue, infact it's pretty easy.
There is a lot more to this tough than to just introduce the two birds together and letting things happen from there ... it will take work, and how it HAS TO START is with YOU! YOU MUST BE SEEN AS THE LEADER OF THE FLOCK in the eyes of all of the members of your flock. So that means not showing favoritism (parrots are smart, they know what they can get away with and what they can't - they are always testing) and being a strong leader ... it's not THEIR FLOCK it's YOURS.
Now, you're choice of a second bird intrigues me greatly ... A hawk head huh, along with your FSIE, why not just bring in a Scarlet Macaw and complete the Holy Hell Trinity of stubborn birds? Hawk Heads are NOTORIOUSLY strong headed and they NEED A VERY STRONG HANDLER, otherwise they become nasty birds very, very quickly. This is a sad situation because when someone spends that much money on a HH they expect them to come trained, and they don't ...
A rare breed indeed the HH, tell me, why a Hawkhead, this isn't an everyday bird that someone says, "Hey, I think I want one of those ... " They are great and awesome to look at and hold, but, man, they can be a handful.