There's a sub-forum regarding CT law that you need to read.
300 BLK does not run better through a 10.5" barrel- it simply performs better than other calibers when shooting subsonic rounds through a short barrel. IDK if you can have a suppressor in CT, but unless you do have one there is no reason to run subsonic rounds. 125 grain 300 BLK loaded to normal (supersonic) specs is a nice short range defensive round.
Unless you have a dog you really don't like and you think you look fabulous in an orange jumpsuit, you need to stop with the SBR activity until you have an approved Form 1 for a SBR. No clue how that works with CT law, thus the recommendation to read that sub-forum and ask CT legal questions there- after you read up.
I strongly suggest you just get a barrel that is either 16" already or a ~14" barrel that can reach 16" with a pinned and welded muzzle device (assuming that's what you need in CT). Assuming you can get a SBR in CT, you'll have to go through the hassle of getting fingerprints, filing and submitting the form, shelling out $200, spending ~$60+ for engraving, plus waiting for approval. Given your budget build oriented questions, doesn't sound like spending the money to file for SBR is a wise decision. Make something that is 16" when finished and spend that $200 on ammo.
A lot of people do it so you are not alone, but burying yourself in over-complication is only getting in the way of you having fun at the range. You should already be shooting something.