Having put a few builds I've done on a "diet", the barrel is where you "feel" most of your weight. Barrel profile is a lot less critical than people think it is with regards to accuracy, particularly within the context of a combat weapon. If you are thinking of buying a new barrel, take a good look at the dimensions of the barrel profile of what you're looking at. I've seen a few barrels that were called lightweight, but in reality were just A2 profile. The M16A1(and Colt SP1) had a barrel diameter of .625" at the gas block, with most of the rest being closer to between .585-.605". Contrary to what most people think, the only reason the military went to the heavier A2 profile was because knuckleheads were bending their barrels trying to open 55 gal drums. Obviously, that is not a concern in your case(I would hope at least).
I believe the HBAR profiles were something born out of the late 1980's- early 1990's. Demand for military type rifles spiked(much like it has in recent "panics")during this time because of all of the rise of the modern anti-gun movement and it's successes. So obviously, manufacturers were going to be looking for ways to increase production rates to meet the demand. an HBAR barrel requires almost half the machine time to turn from a blank as a lightweight A1 profile, because they are generally unprofiled (or profiled very little) under the handguards. This time savings is HUGE when you're trying to make 1000s of something, even more so when you're trying to make 1000s in a hurry. Interestingly enough, when did we start seeing HBAR rifles/carbines? The late 80's-early 90's; right at a point when demand was massive for ARs. Somehow I don't think this is a coincidence. And I'd bet that their introduction had a whole lot more to do with production than it did accuracy.
It may not sound like much, just looking at the dimension change, going from an HBAR to a A1 profile barrel. But the felt difference is HUGE. I had a ban era carbine a while ago that I did this too. It was an HBAR-completely unprofiled (so between .950-1.000")forward of the chamber until the gas block, and .750 forward of that. I turned down the whole barrel forward of the chamber, with the exception of the gas block(I didn't want to have to track down a .625 FSB) to A1 profile. The weight difference? Before I started it weighed more than 9 lbs loaded. When I was done, 6.5 lbs loaded. It felt like it didn't weigh anything at all in comparison to what it did before.
Now there is a tradeoff for this. Thinner barrels are obviously less rigid than thicker barrels are. So you could potentially get some vertical stringing as the barrel heats up. This occurs because of the extra "droop" in the thinner barrel. It's not really a big deal accuracy-wise unless you want a rifle that will shoot 1 MOA or under, even if it's hot. So you should really consider what you want from the rifle. If you want that kind of accuracy, you could still turn your barrel down, but I probably wouldn't take it down as far. You could also flute the barrel. If you're OK with potentially having combat accuracy(which in all seriousness is perfectly adequate for all but long range shooting beyond 600), in return for true light weight than I'd turn it down to A1 profile.