Hey Brian and welcome to the GTA.
I just recently got a Big Kitty, myself and I have to say, although I love this rifle, I've had the same frustration. First off, GET THAT TRIGGER!!! There's no
ifs,
ands or
buts about it. It will go a long way to taming the Cat. You'll need a set of scope rings that won't creep at all so yeah, get one with a stop in it. My suggestion would be the Gamo One-Piece but others have their fave's also. With my Leapers 3-9x40 Mil-Dot scope it seats it nice and low to the barrel. In my opinion, the closer to the barrel the scope is, the better and easier the accuracy will be, but that's a contested theory and up to personal preference.
One of the biggest and best pieces of advice I've learned in my short time being on this forum was: putting enough rounds down range through her until the parts wear in and she settles down. It takes time and I've heard numbers from anywhere between 150 to 1500 rounds before she'll do that. I have noticed that after about 150 through mine that she's coming to heal, slowly but surely. Time will tell.
Since these are also mass-produced weapons, of course the tolerances will be relatively loose on the inner workings. I have heard that a good tuning will do wonders not only for accuracy and ease of shooting, but also for the life of the weapon. As yet I haven't had the chance or resources to find that out for myself but I don't doubt the many testimonies you will find on this site that attest to this fact. Maybe when my stimulus check finally arrives I'll get around to sending her off. As always, YMMV.
As for mods, just for kicks I'll post a pic of mine just to give you some ideas, as I'm sure others will, too. You can also read the short list of the simple mods I've made to mine in my signature, below. I'm sure I'll add more as my tastes refine but for now, she's mine and I'm happy with her. You've got a good rifle on your hands, she just needs some taming, yet.

Good luck and keep shootin'!

LJ
:EDIT:
P.S. also, pointed pellets seem to be erratic by nature for most folks' babies. That may be a good deal of the problem. You're Kitty needs a varied diet until you find a pellet that she likes. I'd suggest wad-cutters or domed pellets to start with as they seem to work for most folks and their "girls". For mine, she used to like the Gamo Rockets, but since I put the bipod on her, she likes the Crosman Premier Hollow Points better. Get a sample pack of different pellets from your favorite website or store and see what she will spit most accurately. Again, good luck!