Is there an EUB with a neck profile of an Engelhardt/Kay? I can only play basses with these necks due to my carpal tunnel syndrome.

I don't know of any, but it would be pretty easy to build your own using a Replacement englehardt neck. They can be had pretty cheaply too. I was going to use one of these for one of my EUB projects, but i decided that huge scroll would look funny sittin on a little EUB.