You can use a brass guide shaped like a little funnel to protect the bore and safely go through the muzzle.
Most of the carbon buildup is right after the chamber and that’s where the brushing must be focused, not the end of the barrel anyway.
I wouldn’t worry too much about copper and lead because they general come out pretty easily, and you’ll quickly leave a new seasoning of building after very few shots anyway.
Each time you disassemble the aluminum receiver 10/22 it further weakens the part where the steel bolt enters the action, so I’d minimize thorough disassembly.
I have always found the entire receiver end to clean up easily using lighter fluid for parts flushing and a brush for the breech and bolt face.
I wouldn’t use a stainless brush on anything. The brass brushes are softer than the bore and the nylon ones are even safer. These brushes themselves won’t damage anything but a misguided rod on the muzzle potentially could. These brushes themselves material that can do damage is stainless brushes, leaving caustic chemicals on too long, and leaving too much carbon buildup near the breech as carbon is very hard.
I run a snake through it after each session, brush and lighter fluid the bolt face and breech when it gets gross, and clean from the muzzle with a guide to protect it when accuracy drops off or when I just feel too perfectionist.
10/22 bores are most forgiving because the bores are slightly looser than that of many others.
Your round count matters less than actual accumulated deposits which your judgment and eyes tell you correctly. I would gauge it by what your choice of ammo is doing rather than a specific number.