I've never dealt with or heard the Avalon U5 live, but it looks like a sweet DI.
We never had effects for bass at our church (all three campuses) either until one day I just showed up with my SansAmp (not my whole pedal board but just the SansAmp) to use in place of our normal DI. I didn't ask, I just did it lol.
During the pre-service warmup/rehearsal, the guitarist stopped mid-song and waved everyone down and said "hold on, the bass sounds distorted." Mind you, it was a song that had some bass OD in the original version.
I locked eyes with him, then motioned my head to the SansAmp and then glanced over at his pedalboard (as if to say "if you can do it, then so can I). He shrugged as if to say "I hear ya, cool." I looked at the WL and she said "let's take it from the top."
I used the SansAmp for the next few times I was in the rotation, then I finally brought in my smaller pedal board (Temple Audio Solo 18) with just a tuner, compressor, EQ, HPF/LPF and SansAmp. The guitarist on rotation that day saw it and said "that is one cool pedalboard!" and asked if I wanted to borrow his HX Stomp to play around with different effects. The rest is history.
I guess the moral of the story is you never know until you try. Unless the WL explicitly dictates he/she doesn't want bass effects, then you have nothing to lose by trying. Even if they did state they don't want bass effects, I'd prayerfully and tactfully challenge them on that...first citing that if it's good enough for guitar then it's good enough for bass, then I'd let them know about the numerous pro worship bassists that use effects. David Curran of Elevation Worship even has a video rundown of his effects rig. Good luck.