I have the same reaction to the real story behind “The Exorcist”; the non-Catholic Christian parents are told by their minister that they need a Catholic exorcist. Could you imagine that? Why return to your regular church the next Sunday?
It's even better than that. The boy's family was Lutheran. (I don't know which Lutheran group, and that might have made a difference.) It is surprising that a Lutheran pastor, of all people, would say, "Here's a tough case; better turn it over to the Catholics." Especially since the Lutheran church recognizes possession and authorizes exorcisms, in principle at least, though it's rare.
I'm not an expert on the case, but I have read up on it from time to time. Some obvious potential explanations come to mind. The case occurred in Cottage City, Maryland, which is opposite Bladensburg on the Anacostia River, sandwiched between the river and the DC line. (William Peter Blatty fictionalized and sensationalized the story, and changed a lot in the process including the location; he chose Georgetown because Blatty had been a student at GU when the case made the newspapers, and he knew the Georgetown area. He also made the child a girl; the real case was a boy, twelve when the events began. The family's house still stands.)
Anyhow, Cottage City was a small trolley line suburb of DC. The boy's father was a federal employee. The local Lutheran pastor and the neighborhood priest would have lived quite close to each other. They would have had overlapping congregations. I imagine that they were friends. And it may be the case that the pastor asked his pal, the Catholic priest, to take a look, since the Catholics do more exorcisms than Lutherans.
I also got the impression, from a source I would have to search to find again, that the Lutheran pastor, like the boy's father, was somewhat skeptical of the whole thing. The true believer in the possession was the boy's mother. It may be that the pastor wasn't convinced, so when the mother was adamant, the pastor called in his buddy the priest for a second opinion to humor the mother. That's all speculative, but it is plausible in terms of small town social dynamics.
I would also not be surprised if the fact that Catholic University is two skips up the road might have played a role. That's Catholic Central, USA. When you have that kind of a resource right at hand, it would be very sensible to hand the case off. That said, I don't know if CUA was ever approached; the university and related seminaries could have sent out whole battalions of experts (who would have very much wanted to consult on such an interesting case), but that apparently didn't happen. That's still a loose end in my book.
The case began in 1949, so there's a substantial mental adjustment needed to try to understand the people of that time.
The family physician had been consulted, but one thing led to another and the boy was admitted to Georgetown University Hospital. That means the docs and shrinks at Georgetown would have had a look at him, and presumably passed their recommendations on an exorcism up their own chain of command. The exorcism was actually begun in the hospital but was suspended when the boy wrenched a bedspring loose from his rollaway hospital cot and stabbed the exorcist. The DC exorcism was suspended. The mom then took the boy to St. Louis, where she had family, and started over. That means a different archdiocese, a different set of priests, docs and psychiatrists, and a different bishop -- a whole second set of eyes, all reaching the same conclusion. That's part of what makes the case so interesting.
We ain't talkin' about an uneducated rural family and Father O'Gullible in the back of beyond, completely isolated and trying to improvise on something above their pay grade.