Even though I have a couple of ARs, My first and only ground up build so far is a LR 308 Xanthos Light weight 6.5 CM build.

I want to start another, less expensive build on the AR-15 platform using a more economical upper/lower combo with a "premium" barrel /BCG

With sets from Aero, PSA, Live Free Armory or Karris Guns out there for a very reasonable cost, if I go with a "premium" barrel (ie Lilja, Proof Research) is it best to keep the BCG brand the same as the barrel, or at least what the barrel manufacturer suggests?

Or since this is all mil spec, am I able to use whatever barrel/BCG combo I want with a certain degree of confidence?

Thanks

Many times with barrel mfrs they will sell you a headspaced bolt with the barrel.

What do you want to use it for? I would use a Faxon, Green Mountain, or other quality barrel, CMC or Triggertech adjustable trigger and bargain shop the residual parts. You should be able to build a .75MOA for less than $850.
Putting a Proof Barrel on lower priced components does not make sense. So unless you are after a match grade bolt and barrel most any milspec bolt will work with a quality barrel.

I mean the Upper and Lower are pretty standard, so it really shouldn't matter.
If you want a matched billet set go ahead, but you're not going to gain much. I bring my upper into my Local Gun Store, and find the lower with the best fit.