Wilkinson used to be a company that made 5-string conversion kits; I had one but sold it on e-bay awhile ago. The ball end anchored on the headstock (device installed on the string tree location) and the bridge had the 5th tuner. Completely retrofitted on Fender basses without any modification to the bass (except for the nut, which also came with it), so it could easily be restored to original stock set-up if you didn't like it. String spacing was tight, however one would eventually need to upgrade pickups for optimum performance.
Obviously, this came out at a time when 5-string basses weren't readily available and the curious didn't want to invest/commit. Personally, I wouldn't bother with the time, effort, and $$$ converting when so many are available. But for your requirements, finding the right 4-string and converting may be the only solution; so one of these Wilkinson kits may be the way to go. They are obsolete and rare, so finding one would be difficult and it wouldn't fit on a contoured top (i.e. hollow-body bass), only flat surfaced bass guitars.