I'm not sure what you are trying to do. If you intend to use an electric bass guitar pickup you are going to get an electric bass guitar sound. If you want the acoustic guitar sound you need an acoustic guitar pickup. If you are playing in a reasonably quiet room the acoustic pickups work OK but if it's on a loud stage you will have feedback problems. If you really need that acoustic sound on a loud stage the best thing to do is mike it with an instrument microphone.
There are underbridge pickups that work pretty good but they require installation. Think it through, What do you want to do?
I know adjectives fail at describing sound, but anyway, I hope I can make a point. Indeed, an electric guitar pickup won't give you a full, electric sound, as the mass of the wood and the absence of frets will affect your tone. I suspect your bass with electric pups will sound "hollow" and not "round" as low-end electric basses do because of their light wood. I totally agree that an electric piezo pickup (I suspect that's what Jeff meant by an acoustic guitar pickup) is the way to go. I have a Chinese made electroacoustic bass by the brand(?) of Freeman(?). When I was trying electro acoustic basses at the store, I tried an Ibanez bass with D.I. output and all, and it sounded really solid and better than the one I bought. I believe the pickup was Fishman, so I think you should look into them.
BTW, Simon, how does the acoustic bass without frets feel? You've made me want to defret mine. It was fairly cheap so I wouldn't mind doing the job myself. I'd need to string it with flatwounds, right?