Having spent several years behind an MP5, and originally trained by Phil Singleton, I can say that there are a few maintenance tools that I would recommend.
First you should make sure you clean the flutes in the fluted chamber. This can be achieved with a good solvent and a chamber brush, making sure that you scrub back & forth in the direction of the flutes.
Second as mentioned above is that the chamber face needs cleaned, as this is where the bolt locks up against it, and if there is a build up it could cause issues. Use a good solvent and chamber face brush. The chamber face brush is available from HK, but you will also find them at E & L Manufacturing https://elmfg.com/store/hk.html, who from my understanding makes the brushes for HK. The mention of a 20 or 12 gauge shotgun brush IMHO would only scrub out to the side where the bristles are pointing, and not forward where the chamber face is, so I would opt for the chamber face brush pictured below.
Chamber Face Brush
MP5 Chamber Brush
E&L as makes a great receiver brush that helps get at all the nooks and crannies of the receiver.
As for solvent & lube, I am a Slip2000 fan and would suggest the Slip2000 Carbon Killer (Carbon Cutter) to help eat the carbon loose. For lube I would suggest Slip2000 "EWL" Extreme Weapons Lube in the standard or "EWL30" for something a little thicker.
Greg Sullivan "Sully"