Verifying the DNA match is definitely a good idea.
Given the option, I'd personally go for fixing the AQHA papers rather than adding the brand. I don't really see brands adding any value in identifying a stolen horse that the registered recorded face/leg markings don't already do. I've seen several of threads in this forum with people looking to trace back their horse's history based on a brand and never seen it be useful (except in the case of JC lip tattoos or mustang neck freeze brands, but that's because of the central data repository associated with them)
“The horse is a mirror to your soul. Sometimes you might not like what you see. Sometimes you will.” - Buck Brannaman
"Nothing forced can ever be beautiful." - Xenophon