You may not have to apply anything after this but if you do, consider a light film of choke tube lube as it's rated for a higher temp. By rights, you should not have had this much trouble in the first place. .....
why did you want to take the barrel off?
Why should we ever want to pull a barrel on a 10/22? It's pretty obvious as we like to tinkler. Not so much now, but I use to pull and put them on quite often. ....
Be Safe !!!