Looking good so far. If you don't have molybdenum disulfide grease you should get some for your barrel. AeroShell Grease 64 (Formerly 33Ms) is MilSpec MIL-G-21164D and a 14 ounce tube should be enough for dozens of builds. It takes so little to do a barrel nut and upper reciever, a receiver extension (buffer tube) and lower, and flash hider, that I've been greasing the ball on my trailer hitch with leftover grease for 20 years. Moly is a lot better than lithium grease under pressure and heat. It's $22.84 at Graco Roberts and may be cheaper somewhere else. I only looked 3 places. It was almost 31 bucks on Amazon. I used a different brand I bought at a local bearing supply or spring and suspension shop. It met military specification MIL-G-21164. GREASE, MOLYBDENUM DISULFIDE (81349). NSN 9150-00-754-2595. It was so long ago I don't remember who made it but it may have been Dow Corning. I emptied the tube and what I have left is in a small bowl I dip onto the trailer ball.
Diester liquid lubricant, gelling agent and molybdenum disulfide. High temperature and low evaporation and oxidation resistant and wear resistant. Effective Temperature Range Rating: -100.0/+250.0 degrees Fahrenheit. Minimum Dropping Point: 329.0 degrees Fahrenheit.
https://www.gracoroberts.com/lubricants/aeroshell-grease-64-variant/ AeroShell Grease 64
An Extreme pressure Airframe Grease
AeroShell Grease 64 (formerly AeroShell 33MS) is an EP airframe grease based on the proven lithium thickener technology of AeroShell Grease 33. It can help to extend component life through its exceptional load-carrying and EP properties from fortification using 5% molybdenum disulphide and its excellent wear and corrosion protection. It also helps to reduce the risk of product misapplication by being fully compatible with AeroShell Grease 33. Together, these greases are perfect partners, thereby helping to reduce costs by protecting over 98% of the grease application points on a typical airliner. AeroShell Grease 64 is fully approved to MIL-G-21164D.
https://www.shell.com/business-customers/aviation/aeroshell/aeroshell-greases/about.html