Welcome to the forum, kat, Gigi and Gizmo. I have a 5.5 lb chihuahua mix named Gigi, too (I call her Gigiolina Topolina - topolina meaning female mouse in Italian

) and a cat named Gizmo (he is arriving tomorrow after being abandoned by a neighbor when he moved out).
Now, you are operating under a misconception. It is only bonded pairs that should not be separated and your birds are not bonded. For one thing, three months is way too short a period of time for the birds to bond, it takes them longer than that and the process should never be 'expedited'. They need to make up their own minds about it. And that's why you are having this problem and why they fight - which is NEVER normal for bonded pairs, mind you. Bonded birds never fight. They might squabble a bit over a piece of favorite food (usually the female trying to steal it from the male) but they never fight. Not ever.
For what you tell us, I am suspecting that Gigi was kept a human schedule and free-fed protein food which made her overly hormonal - thus, her insistence on being close to Gizmo. But, because Gizmo is not yet bonded to Gigi, it doesn't want to be with Gigi all the time and resents her insistence.
The solution is easy: you need to put them in separate cages, wait for Gigi to 'lose' the extra hormones and for them to bond to each other before you put them together (they will pretty much do it on their own). Now, Gigi will not stop producing sexual hormones until the fall and then ONLY if you keep them to a strict solar schedule with full exposure to dawn and dusk (make sure it's a whole two hour period because it makes the process faster when you start off with an overly hormonal bird -turning off the light at night does not work because it does not allow for the gradual increase in melatonin as the light decreases or for the special twilight that turns on or off their internal clock) and do not free-feed protein food. Gizmo wants to go with you at night because he is not sleepy and he knows that going into his cage means he is going to have to put up with Insistent Gigi.
When you have a bird that has lived with you alone for some time and has bonded with you, it takes longer for him/her to bond with another bird because it has to 'wean' itself from human attention and learn to enjoy bird attention - and rushing things only makes things harder for both the birds and the humans. It's best to allow them to set their own pace and choices and just observe the process to make sure it's 'pointing' in the right direction. You need to keep the cages very close so they get used to hearing and seeing each other, let them both out at the same time and encourage their interaction with each other, pay extra attention to the 'old' bird while beginning to bond with the 'new' one and, very gradually and very slowly, move toward treating them both equally (and sometimes, you will have to take one step back to your two forward on this so as not to hurt Gizmo's feelings).
You need to look at this process from their point of view. Parrots are not like canaries that would breed with any other canary that is of the opposite gender... they are more like little people and do not like to be imposed on - not by people and not by other birds. They choose who to love, when and how. And they always - ALWAYS- take their own sweet time to decide. I bet that poor Gizmo is very confused right now. You brought an intruder to the home and stuck it in his cage (and you said he was cage-protective so, in reality, you were lucky he did not outright attack poor Gigi) and do not seem to be 'listening' to him when he is showing you as best he can that he does NOT want Gigi there. Be patient, put them in separate cages and go back to square one and, in time and if you do things right (light schedule, diet, etc), it will work out perfectly.