Stay away from a Jag altogether. They can be kind of a pain unless you get the patch thickness just right. (heck I stuck one good myself once!) Use either your idea of a swab wrapped around a .45 cal brush (I have done this myself) OR the easiest is to get a 16 gauge shotgun bore mop on another rod kept nearby.
The bore mops work very well, never get stuck and stay moistened with your swab liquid. I have several and use them for swabbing while shooting a well as final bore protective swabbing when finished cleaning. Easy to clean, a little dish soap, finger scrubbed, rinsed, squeezed and set out to dry.
Lots of between shots bore swab liquid recommendations but a good 'old school' swab is 50/50 Windex and Murphy's Oil Soap.
Bore mop.
Ok well there's my Homer Simpson moment....D'OH! I have a bunch of bore mops in various sizes and even bought a .50 cal one for this rifle. Thks