A Master Electrician here to help. I am a double qualified; USA & UK electrician. Please give me a moment to formulate a reply.
Unless you have documentation to show that Siemens states these Murray breakers are an acceptable alternative to their own both as a breaker and for series rating then any electrical Inspector caching it is likely to red tag the inspection.
I suggest you contact your local inspector to ask his view.
One other point here;
Your barn being 240 ft away and with a 200 amp feed would require 4/0 copper which has a resistance of 0.0608 ohms per 1,000 ft. With these values then there would be an available fault current at your barn of 8,218 amps.
So even if we thought of the AIC as being infinite before your 10 KAIC breaker then still only 8.218 kAIC is available at your barn.
If I include an AIC not exceeding 22 kAIC as a feed from the main for this circuit then the final available fault current at the barn will not exceed = 5.985 kAIC
Now this figure is "at the barn" of course, if there were to be a fault in the 240 ft of wiring to that barn then the fault current would increase, but the most likely location of a fault would surely be at the barn unless later excavation work is done damaging that feed.
I hope this helps.
I hope this is simpler;
Even IF series rating does not apply then in any case;
Your main breaker is rated at 22 kAIC and your barn breaker is rated at 10 kAIC. When In fact your barn service AFTER the 240 ft run only needs a minimum of about 6 kAIC.
This doesn't make it right because a fault could occur between the barn breaker and the 240 ft of wire at a point where there might be MORE than 10 kAIC. But it makes it better.
(Higher the kAIC rating the better)
You CAN use my calculations to show that no matter what the case ALL breakers in the barn panel (s) will only need to be 10 kAIC (the minimum normally available).