Using a pull-through unaided on ANY bore is near-certain destruction of its accuracy. A stainless rod is considered by many to be the BEST rod for prized bores. The Brits in WWI in addition to a pull-through issued a device that clamped on the muzzle of a SMLE and protected the precious last few inches of the bore. Look around on the net. They're about $50, but well-made and essential to protect the bore. You'll have to enlarge the part that clamps, as the Enfield's muzzle is much slimmer than most BPCRs. Actually, a scratched stainless rod is no problem, little threat to a fine bore.

If I am not mistaken, CPA makes a hammer that is designed to allow clearance, but it may be an extra cost. If your hammer does not clear, you could talk to them about getting the one that does clear. Probably the simplest solution.

I have used the same pull through I got with my Anschutz .22 25 plus years ago.
it it nothing but a coiling nylon wrapped deal with a loop for patches. The patches provided look like old mop strands. I've not had any accuracy issues or chamber or crown damage. And I clean my 22 rifles once a year if they  need it or not!

I've used self made ones on match airguns s well for longer than i can recall .these just pull patches nothing more.

I see no issue with thus type of pull through.
Anschutz still sells them so I guess they don't either.

I use a .17 caliber stainless cleaning rod for my .22 rifles. Being a little smaller diameter makes less chance of contact between the rod and the bore, and it's worked well for my rifles.

I personally wont ever use a brush of any kind on my .22 bores. Some solvent and a patch does it well, and is less abrasive to my bores.

I quit using wire brushes after one visit to a gun smith named Richard(Dick) Hart in Boise, about forty years ago.  He took a new copper wire brush and drug it across the outside of a polished piece of stainless barre.  He then handed it to me in one hand, and a 10X magnifying glass in the other.
The nylon ones these days are so much softer.  Some people do not think it affects accuracy, but my last RKS gain twist barrel was four months and five-hundred dollars; and did not include the fit, or chambering, or head spacing or bluing.

My idea of gambling involves a full roll of nickels at a Casino.

Rich

My two Douglas barreled CPA .22 barrels either one in service over 20 years, don’t need much cleaning. Light spray with Balistiol let it sit then pull through with a single oiled .22 patch followed by another clean patch  . Pull through home made,  loop tied in 50# monofilament leader material. It’s stiff enough to push end through from the breech so it can be pulled out the muzzle. Anschutz Walther & all the top Air Rifle makers supply pull through with the new gun.

Use the same routine with a stainless rod. Rod can contact the extractor but it’s only making a couple of passes, keep it away with my thumb. Pull through or rod depends on cleaning at the range or home. Barrel on the gun or off when it’s switched to CF

I never scrub “good” 22 barrels with a brush. Old and rough yes, new Button Rifled Douglas does not need it. 
As mentioned above Delrin rod good solution, that’s how I wipe my CF barrels. .22 Delrin needs to be so small I prefer the stainless rod.

My guess since non corrosive ammo came out more barrels ruined by over aggressive cleaning than lack of cleaning.

Boats