In my Thompson Center Black Diamond XR there are a number of loads that shoot excellent. Most of the Thompson Center projectiles you purchase that come with bullet and sabot will shoot excellent. If you are going to go outside the T/C line of bullets then get some Thompson Center Mag Express Sabots. It will help on the loading. You still need to swab between shots for the greatest accuracy.
Make a swabbing solution of 50/50 isopropyl alcohol and car windshield washer fluid or windex. Dampen the patch good. And work it up and down the barrel in short strokes from top to bottom so it does not become lodged in the barrel. You can follow up with a dry patch if you like, but again, short strokes.
My rifle shoots the 250 & 300 grain shockwaves excellent. It does shoot the 250 grains better but you would not want to live on the difference. I shoot 100 grains of Goex FFFg. I have pushed these to 120 grains and the accuracy was still very good. The two 777 pellets should be fine.
In the Thompson Center Mag Express 240 grain XTP you might want to back off the 100 grain charge a little. If you push them too hard, at longer distances you get flyers. I shoot them with 85 grains of Goex FFFg.
An excellent hard hitting bullet is the Buffalo Bullet 375 grain SSB. They are a spitzer hollow point. I get them through Cabela's. I shoot them with 100 grains of Goex FFFg and the accuracy is hard to beat.

This was an end of the day shooting experience. I had shot off almost 30 rounds and was tired. These groups were shot at 100 yards with the SSB's.
If you want to go to the full size conical and stop anything that moves, then get some
No Excuses Conicals. These monsters weight 460 grains and that pack a punch no doubt. I was shooting them with only 80 grains of Goex FFFg. They are easy to load and super accurate. The best group I got with them was shooting 80 grains of Triple Se7en. At 100 yards they shot a big nasty hole through a pine board.
You have a great rifle there. In order to keep you from having some of the learning problems I had, get yourself a 8" extension for your socket set. Put a 7/16th socket on the end and it is the best and fastest breech plug remover there is. Also get some CVA Slick Breech Plug and Nipple Grease. Use it. It is the best. That stuff from Thompson Center is all right but you can't trust it. The first time I used it I got a stuck breech plug. When shooting on the range, after seven shots, loosen the breech plug with that wrench that came with your rifle. Just work it back and fourth a little. That will keep the breech plug from getting stuck.
I have the same scope you have on your rifle, a Bushnell Banner Dusk to Dawn in 1/5 X 4.5 power with 3.5 inch eye relief. I hope you got at least that kind of eye relief because this rifle will catch you sometimes.....
Do yourself a favor and save some money at the same time and get loose Triple Se7en. It can be tuned better and is cheaper in the long run. If you insist on running pellets pick up some of the 30 grain also so you can make some of the different sized loads.....
Any questions you might have, just ask. Any way I can help, just ask. Good luck with your rifle..