If you are going to use cast, you will have to slug your bore to know what size bullets you need. I spent the best part of 2 years tryig to get my 44 to shoot cast good, it turned out that my groove diameter is .432 As soon as I stopped sizing my bullets to .430 and left 'em as cast, my groups size shrank by nearly 3/4" at 50 yards, and I was able to push my 265 plain base bullet above 1400 fps accurately.
The only 300 grain cast bullet I have used that will stabilize in the 1:38 twist of the Handi is the Magna Bullet Co. 300 grainer (#704 IIRC) I have tried almost every 300 grainer out there, this is the only one that didn't keyhole. 18 grians of H110 pushed it to 1435 fps, max load is 21 grains, IIRC.
The Hornady 300 XTP is a fine shooting bullet. 21 grains of H110 pushes it 1525 from my rifle. the Speer 270 Gold Dot is an exellent bullet too, the max charge of H110 pushes it at 1575 25 grians of 1680 does 1340 with this bullet.
I would stay with bullets of around 270 grains. They seem to offer the best balance of velocity and retained energy at the ranges you speak of.
Saeco makes a nice 265 grian bullet in both a plain base and gas check design that is on the top of my "I want one of those" list if you are into casting your own. I've not had good luck with the classic Keith designs, but that cold be more a result of the bullet size than the bullets themselves.
I throughly enjoy shooting 7 or 8.3 grains of Blue Dot and any of the 200 grain Cowboy Action Bullets. This load produces about 800 fps, is quite, cheap and you can shoot it all day without discomfort. 5 grains of AA#2 or WW231 will get you there too. I prolly shoot a couple thousand of the Blue Dot load every year. Oh yeah, light the Blue Dot with Federal's Large Pistol Magnum primer, they helped my groups a lot. I can help you with some more, but this will keep you busy for at least a year :grin: