PennyPhoenix, I use a Dewey 1-piece brass cleaning rod with a bore guide on all my 22 rifles and find it works very well. The flash suppressor won't make any difference ... in fact it might work even better because the hole in the suppressor is slightly larger than the bore diameter and allows the cone to seat better. The bore guide is nothing more than a brass cone with a hole in the center ... just big enough where the rod will slip through with minimal play. When the bore guide (cone) is held in place with one hand as you are pushing / pulling the cleaning rod with the other hand, it keeps the bore brush centered and keeps the rod from touching the crown.
Some people like bore snakes ... some don't. I'm a "don't". Bore snakes do a good job of cleaning powder residue but in my experience, they do a poor job of removing lead deposits. I much prefer to use a bronze bristle brush for scrubbing stubborn lead deposits. To each his own ... as they say.
There is a very simple modification you can do with any 10/22 that is much better than a front bore guide. It requires drilling a 1/4" hole in the back of the receiver. When the receiver is mounted in a stock, the hole is not visible. The hole acts as a bore guide and allows you to clean the bore from the chamber end. This does not affect the function or strength of the receiver but it does require tearing your 10/22 down and removing the bolt before cleaning the bore. I do a thorough cleaning of the receiver and bolt every time I clean my 10/22s so it doesn't add any inconvenience for me. I did this modification on my "build" but my other 10/22s are still cleaned from the muzzle using a Dewey bore guide.