Good evening, I'm Dr. Sara. I'm a licensed veterinarian who works exclusively with cats and dogs. I'm sorry to hear that Lyla is having a difficult time adjusting to your new kitty. While I would certainly suspect that this is a behavioral issue, it always is a good idea to discuss and check for medical problems before making saying "behavioral" for sure.
There are a couple of things I consider when I have cats who are missing hair or licking areas of their fur out. Number one is itchiness, which is usually allergy related. If the problem is allergy related, then it would likely improve with steroid treatment but relapse and become worse again when the steroids wear off. I have had great success in maintaining allergic cats with a medication called Atopica for cats. It is a daily to every other day maintenance medication to control allergy symptoms. Make sure that she is absolutely FLEA FREE, as fleas are a very common allergy trigger - even ONE bite can make her absolutely miserable. I'd be happy to talk more about fleas if you'd like - just let me know. In rare cases itching can be a symptom of a mite infection, but this is pretty unlikely.
The other thing I consider for cats who are losing their fur from over grooming is anxiety or stress. Cats calm themselves by grooming, so if they are anxious, some cats can over groom and pull out their hair like this. In these cases, the hair usually doesn't grow back with steroid treatment because the problem isn't an allergy. These cats can respond well to treatment with anti-anxiety medications (I usually start with fluoxetine). It's often difficult to determine exactly what is causing the anxiety - just like some people who have anxiety or panic attacks can't always tell you exactly what it is that they are upset about. In your case, I think we have a pretty good idea why she is anxious.
If the hair loss is mostly over the belly, another thing to consider would be bladder disease. If a cat is over grooming their belly and pulling out the hair there, I also always check for bladder problems. If their bladder is painful because of an undiagnosed bladder infection or bladder stones, then this could be the cause of the over grooming.
Here is also a link with some more information on over grooming and hair loss in kitties:
I would have him looked at by your vet as soon as you can. While it's unlikely to be an emergency, if she is really itchy, she will appreciate relief ASAP. This should give you some ideas to discuss with your vet for your kitty's care going forward. Please let me know if I can answer any other specific questions for you!