While musical theater is my main gig, I have also been playing in cover bands - mostly pop, rock and jazz (when doing corporate, banquet type gigs) - for over 50 years. I played most of that on a 4-string ‘65 Jazz and a 4-string Fender A/E fretless. I put a HipShot Detuner on the Jazz maybe 20 years ago. Around that time I also acquired a Roscoe Beck 5 and a Turner Renaissance fretless 5. More and more of my theater and cover band work began calling for 5-string bass. Like the OP, I just preferred playing the 4-strings. For a while I would switch to a 5-string when necessary, but soon realized that it was just easier to play the 5s all night. In the cover bands even 15 seconds to change can seem like an eternity when the dance floor is packed. Playing musical theater taught me to be comfortable playing in all the non-rock keys. The show that I’m currently playing has multiple tunes in 5 and 6 flats.
Admittedly, I don’t play with many guitarists who consistently play in down-tunings, with Superstition in Eb being a notable exception. I also don’t play much music that was written in lowered tunings. However, I do play with a lot of guitarists who use capos and the occasional open tunings. Over the years I’ve learned to trust my ears more than my eyes when following a guitarist who plays in a non-standard tuning.
With all that said, in my experience, playing a 5 string covers just about everything I need. Yes, there are times when I have to really stretch to play a low note pedal and higher note fills, or alternatively I will switch octaves, or just do what I can to make it work.
What surprises me with the OP’s situation, is that around here and in my experience, with cover bands, the vast majority of the audience doesn’t care if you play every note and every lick exactly like the record. As a matter of fact, it is kind of expected that the band will have their own take on songs. Otherwise, the audience can listen to the recorded music in the background. Certainly, for much rock it was non-idiomatic to play live exactly like it was recorded; pop music was maybe more so. It might be different for other genres. Unless one is playing in a tribute band, where note for note accuracy is expected, I just don’t see where a basic 5-string in standard tuning wouldn’t fulfill the needs of a cover band.
Good luck with your new gig!