I understand that the reason you do not want copper remover to remain in your bore too long is due to the ammonia.   Can I remove the ammonia with brake cleaner, or do I need to use rubbing alcohol?

Thank you very much.

I always used shoters choice solvent to remove. 3 to 4 patches after dry patching to lower the molarity enough to where it would not be a problem.

If you flushing with brake cleaner after dry patching, I see no problems.

plain old distilled water would work.

There should be no problem with the brake cleaner as long as its not chlorinated such as trichlor or tetrachlor.Chlorinated solvents and ammonia make acid when mixed.Very nasty for the bore.Personally I do not think its the way to go or effective.To be honest I have no use for the spray solvents on weapons,but thats my opinion.YMMV
I'd also not use any of the other solvents in conjunction with the ammonia based solvents.Ive read several times on the net that the shooters choice bore solvent acts badly with Sweets7.62 resulting in barrel damage.Enough for me to be nervous about it anyway.
In fact its a good idea to just not mix bore solvents.
Personally I use 91% isopropyl rubbing alcohol to flush the bore before using a different solvent or when Im done with the ammonia based solvents etc..
The easy to find 70% isopropyl alcohol works fine as well but it contains a higher water content and does not flash off as well.I use the 91% due to the fact I keep a supply here to use for mixing the TW-25B EP spray.
One or two patches of the alcohol will do it then dry patch and oil immediately.
Key to remember is the ammonium salts are hygroscopic/water soluable...so thats the best way to flush them out.That said the isopropyl alcohol contains water which is enough to remove the trace amount of ammonia remaining,and the alcohol causes flash evaporation of the water.Plus the alcohol removes all traces of solvent leaving no residue that could affect the lubricant used in the bore as a protectant.Just remember to oil the bore immediately when your finished.

+1 on 91% isopropal alcohol. It is also non corrosive. I left some shiny iron in it over night and no rust ever formed.