I always look to what the client is asking for. That means I refer to chiropractors, MT's, acupuncturists, etc. If the client goes elsewhere and gets the relief they need/or the additional care they are seeking (even if they don't make it back to your doorstep), then you have done your job.
I have a chiropractor I refer people to, who takes insurance, since I don't. If someone needs that kind of thing, my main concern is that I refer to someone who is good at their work. What I don't want to do is refer a person any practitioner who has crappy skills.
If they like your work, they'll be back. And, sometimes if they like you work, they won't be back (if they find someone else who is similar and can do the same job). And, that's ok.
Client requests come first for me, and I believe that they appreciate me and recall me in the future as an honest person, even if they end up staying with someone else for a while.