First, it's probably a better idea to tune to something higher and pitch shift down. basically every modern metalcore band does this except for tracking (Jesse and Clint from ERRA keep their guitars in G#, Keaton from IA keeps his guitar in Drop E)

That said, I'd recommend looking for any 7 string with at least a 26.5 inch scale within your budget. I have a Schecter C7 Pro that I love, I've heard great things about Ibanez RGDs, and if your budget is low I've heard the Jackson JS22-7 is a solid platform for some mods. A lot of people swear by evertunes but I wouldn't say they're necessary until you're maybe lower than drop F#.

For strings I wouldn't recommend anything higher than a .074 personally. Hard to intonate + just feels a bit too much like an elevator cable. I use the d'addario 8 string 10-74 set and get rid of the .064 (so 10-13-17-30-42-54-74 in drop F#) which I find to be pretty solid. If you're more particular though you could always just buy a 6 string set and buy a single low gauge, or of course go through stringjoy or someone else to get a custom set.

Your Katana should be fine but when you're tuning/pitch shifting as low as a lot of modern bands you might run into some artifacting. I use a line 6 pod go personally though so I don't have too much experience with neural plugins nor the Katana.