Everywhere I've known that has the rule of no camera (such as my mother's place of work, for the local council), a) provide the handset and b) disable the camera if no camera-less alternative is available.
This is a really common request for those who find this unusual - cameraphones are banned in a lot of industries but the main ones are security, gov work, aerospace and defense, R&D etc...
One trait is exceptionally common though - 99% of the time these phones are only available on Business Tariffs through full business accounts. I'm not sure you'll be able to acquire one through usual channels unless you're looking for a dumbphone.
Surely any organisation with this rule should at least be able to give you a list of approved handsets even if they won't provide one. Have you not asked any of your colleagues (or future colleagues) what they use?