Push 1-2 DRY patches through the bore with the breech plug out. This will get out a huge amount of fouling without making a mud pie.
Clean with whatever centerfire solvent you like. Even good old Hoppes #9
Ditch the BoreBummer
Use any quality synthetic or mineral oil based storage oil. Ballistol is cheap and easy to find if you want a CLP. Its main ingredient is light medical grade mineral oil. Its compatible with all powders too.
Avoid oils with Teflon added. There is NO REASON to introduce Teflon into your bore.
No matter what oil you use its best to swab it out before you shoot again. Since i do that i use WD40 Specialist Long Term Corrosion Inhibitor....This AINT regular WD40 but it also aint a synthetic. Its is however a great oil for storage.
A little bit of brake cleaner on a patch removes bore oils fast and its cheap. Just keep it off stocks.