I too wonder if Lydia is aggressive or playful. She may well need an active playmate to take out her energies on so Briggs can live out her life with a little peace. (This exact strategy worked for Payitforward and her 1st, 2nd and then 3rd cat).

If it is an agressive thing, medicate her. Talk to your vet. I house a lot of fosters, and I have one right now who is playful/dominant . If he attempt to play (and play rough) with a very shy cat who runs from him, he tends to intimidate that shier cat. I gave him a dose of an anti stress drug that I use for 2 of my own cats. It seemed to do the trick, and I only had to use it for about 3 days until a new behavior was established.

Don't give up after only a month. It may seem LONG while you are living it, but really in the scheme of things, try to give it more time.