A relative of mine, who was still officially a baptised Jehovah's Witness though had slipped out the back door of the religion for many years, came very close to getting the help of a Catholic priest to do an exorcism. Not for herself, but for a work colleague who appeared to have become demon possessed after an ouija board session. This was in a hotel where she and a group of fellow workers were on a course and, one evening, they played with the ouija board, for a laugh.
It turned into a frightening experience later on, so awful, in fact, that their fears for this chap were such that she grabbed a telephone directory and started looking for someone who did exorcisms to come out in the middle of the night to help deliver him. She expected there would be Catholic priests under that category, but she could not care less who did the exorcism, just as long as some person responded to the call for urgent help. As it turned out, the chap came back to his normal self before she could phone anyone. Everyone went back to bed, but she woke up later with a palpable sense of evil in her room. She said the whole event was terrifying.
However, although she was technically still a JW, and has never been disfellowshipped, she had long stopped being a JW and had no contact with any Kingdom Hall. I do wonder, though, if she had been known to JWs and they got to hear of that event, whether she might have been called before the elders to explain herself. As one JW has already commented here,
"Given the fact that a JW severs all ties to any other religion before being baptized it would be viewed as a serious sin to seek out
spiritual assistance from another denomination in this way."
When JWs speak of 'serious sins', that usually entails examination by the local elders, with a view to discipline or even being disfellowshipped. For that reason, you may be confident that no practicing JW would ever admit in public to seeking Catholic exorcism.
It's hardly like that for Baptists. If a Baptist sought exorcism for anyone, they would, in the first instance, seek help from mature Baptist leaders, naturally enough. But if for any reason they called upon Catholic help (suppose no mature Baptists could be contacted, say) and this became known, they might be asked to explain, but if the outcome was relief for the sufferer and nothing else transpired, fair enough. I guess, however, that if anybody involved became a Catholic as a result of the exorcism, someone's name might be removed from the Baptist register of members. But, I've never heard of such a thing in all the years I was on a local Baptist church register. And I don't have access to ministerial instruction books for such events! You really need to get a Baptist minister to answer this.