I ran into this problem as well and I really wasn't sure how to approach it. I play in two bands, a cover band that tunes to Eb, and now this original band which tuned to dropped B. I only have so many basses, so to have different setups on them for different bands is tough. I first considered dropping my 4 down and having it setup accordingly because I like playing a 4-string better. After thinking about it I decided that probably wasn't the best idea. Like you I also have a 5-string and was considering staying in standard tuning. That posed a problem for me because there are alot of fast, open chord riffs that are 10x easier to play if I'm in the correct tuning as the guitars. I don't really want to be all over the fretboard just to play a simple riff.
What I do now is just leave the low B as is and tune up the other strings. I initially didn't want to do this but after having it done and adjusting the bass, it plays great and it sounds really good. I'd suggest giving that a try assuming your guys tune like we do...
B
F#
B
E
G#