Been thinking about this, even though I don't have a Cross, yet...
How about wrapping a bore mop in a circle around the barrel, or part of a circle if it's not long enough, then twisting the threaded end 90 degrees and using a cleaning rod to slide it from the muzzle towards the receiver?
GLWS. Every time I look through mine I'm like WOW, that's sharp. Seems to be the easiest scope I own (Leupolds, Meoptas, other lines of Vortex) to get a bright, clear, no edge shadow picture. Compact and light for its magnification power.
Rimfire Central would be a good place to advertise this classic. You're in New Jersey Steve? The picture of me and the pup was taken on the wing dam just down river from Lambertville.