I recently picked up a used (and kind of beat up) Audio Technica Pro 95. This is a clip-on gooseneck cardioid mic originally intended for guitar, and was actually designed to be mounted inside the guitar body. The frequency response is from 80-13,000Hz. The mic is powered by a tiny in-line pre-amp with a watch battery and the cord terminates in a standard mono 1/4" jack, which makes it convenient for those without phantom power or an XLR input.
Anyway, I didn't have high hopes, but I got it really cheap so I figured I'd try it out. I have to say, I was very impressed. The tone isn't the greatest I have ever heard, but it is a whole lot better than just a pickup (I usually mix it). Most importantly, in my view, I find it almost impossible to make it feed back, even running it directly into my amp right next to me on stage. This has been my #1 complaint with almost every single mic I have used live--you slowly turn up the volume and just when you start getting to the point where you are getting enough gain to make the mic worthwhile, you start getting something annoying: screeching, low booming notes, strange ringing sounds, or whatever. With this mic I am not having this problem. I have been able to do some gigs with the mic only that I didn't think I would ever be able to do that on.
Unfortunately, this mic appears to be discontinued, but if you happen to find a cheap used one that you can try out, give it a listen. A couple of little issues I have with it: first, I occasionally got a bit of low-end boominess when I played the E string. I think this was mechanical noise transferred through the tailpiece where I have the mic clipped on. I fiddled with EQ and placement and it is better now. Second, the gain is quite low ( in spite of the manual's claims to the contrary). I have to crank the pre-amp level on my Clarus way higher than I ever had with another mic or pickup. This could be a problem with this individual mic (it is used and kind of beat up).
Anyway, the moral of my story is that sometimes the ideal mic for your bass may not be one that is normally thought of as a bass mic. Try as many as you can.