Ok here's my opinion. I have one son via IVF (frozen cycle) after a few IUI's with injectables. My son is almost 2 and we haven't gotten pregnant on our own. No official dx, only that he's got lowish (normal) numbers.
I don't feel like I can do the IVF right now but I have come to terms with the notion that I have to do that to continue to grow our family (and to give our frozen embryos a chance). So I have come to terms with that, mentally preparing over the next several months. I am not going to focus on getting pregnant by chance or by IUI which hasn't worked for us.
If I were in your shoes, I would find a doctor that would be receptive to my wishes. Don't think they have to drive your treatment plan--make sure there's a give and take. Tell them you want to do one cycle you feel confident that's the route, but be open to the possibility that IVF is in your future and spend some time thinking about whether you want to spent your time and emotional energy on something that hasn't worked in the past. Often failure will drive you to want something more and you may find that it makes you want to do IVF again--and if you can arrive at that conclusion on your own, without the expense of an IUI that doesn't work? Then you're two steps ahead of the game.
That said, if you want to just do an IUI without injections and you are comfortable with that being your "one and done" then you should get a doctor to agree to that. You're not asking much. Just meet them halfway if they want to do a few tests to make sure that nothing has changed after having the babies.