The real problem with graphite isn't the "galvanic corrosion" bugaboo. Yes that happens - in other settings. Loose graphite will draw moisture out of the air, and if it's between dissimilar metals, it forms an electrolyte and facilitates galvanic corrosion. But grease with graphite shouldn't let it happen because the graphite is so coated by the grease that it can't draw moisture into it.
What you really need to be aware of is that graphite is supposed to add to the lubricating quality of a grease, but in a barrel nut application the structure of graphite doesn't really fit. It can "scrub" either the nut or the upper, while reducing lubricity. That lubricating quality is what's needed to apply torque evenly.
I have about 14.0 oz left in my 14.5 oz cartridge of Aeroshell 33MS, so I'm pretty much set for barrel nut grease. But if I didn't have that, knowing what I do now, I'd get a good quality assembly grease like MolyKote G-N (from Dow). You aren't going to be getting to 45,000 feet altitude, and you're not going to have to pass a military QC inspection, so using a MIL-SPEC (actually MIL-G-21164) grease isn't absolutely critical. Just use something that isn't cheap crap.