So my current build is based primarily around explosives, grenades and mines in particular, and I want to back it up with guns, as explosives aren't always appropriate (in lonesome road I always feel like I'm blowing away valuable loot every time I used red glare) and since I have a luck of 1, I want something automatic to clear the remainders from when my grenade leaves a survivor or two. I'm thinking the assault carbine or .45 SMG... thoughts?

LMG, hands down, for the following reasons:

AP ammo. Sometimes the thing you'll want to finish off will be a Deathclaw or a Radscorpion, and you will need that DT reduction. Neither 12mm SMG nor the 45mm SMG is any helpful in the AP department.

Very large magazine capacity means you will rarely be reloading mid-battle.

Lighter and less ST-demanding than a minigun.

Assault carbine has a lot going for it. It's ammunition is common and inexpensive and even the base line version has -10 to damage threshold. The armor piercing ammo is -25. Downside is it's total dps is decent but not overwhelming and the mods for the GRA version can be hit and miss to come by. If you do have GRA the gun can be bought from Level 1 at the Gunner Runner's main branch as soon as you can save up the money.

.45 SMG is available as soon as you're ready to take on the Honest Hearts expansion. Joshua Graham will always have the mods for it for sale and, once you're back in the Mojave, so will the Gun Runners. With the Grunt perk this thing is a real beast for dps but loses some of it's punch against high damage threshold targets. That said, between the -4 DT from the Super Hand Load ammunition (you need the Handloader perk to make it) and the 20% bleed through it's dps is probably still higher than the Assault Carbine. Downside is that it and it's ammo is heavy if you're playing hard core.

12.7 SMG has excellent baseline dps which easily surpasses the .45 smg unless you have the Grunt perk, which the .45 benefits from but the 12.7 doesn't, in which case the difference between the two is negligible. It comes along fairly late in the game and the mods and ammunition can be hard to come by. If you just want it to finish critters off with though ammo shouldn't be an issue.

LMG's are heavy and only start dropping as loot from super mutants later in the game . I haven't worked with them a lot so I couldn't really give an opinion on them but their damage doesn't really seem to justify their weight to me.

If, as you say, you just want a weapon to finish off critters with crippled legs you might also want to consider a Hunting Shotgun. They have excellent one shot damage and, if you use slugs, they can also be decent for punching through damage threshold.