Note: all bass players run straight into a DI
I'm no scientist, but I play one on TV
Nah, actually as a sound tech, I'm pretty pedestrian, though I've been doing audio of all sorts for a lot of years. Never had such a problem. No matter what bass.
#1
I think a lot of musicians, yeah, bassists count

tend to crank volume based on what they/we are hearing from where we stand. If everything goes to 11 on our instruments, it's because we're not hearing ourselves well. And to be sure, in big rooms like churches, it's super easy to have lo freqs over play the room's ability to keep it clear. More volume isn't the answer, but to the player, where they stand, they're max'ing it because they're not hearing clearly. Make sure their speakers are within line of ear for them, that'll help them not max things out as a reflex to "I can't hear enough me". But be sure their line of ear isn't killing someone else across stage, and getting into microphones, a good idea anyway, which might be part of your problem bass bleed into mics.
#2
Regarding not getting good sound for every bass via the PA, that's really odd. The first thing I'd want to do....
[btw - do you have any similar problems with any other instruments? real guitars, for example, aside form those wanna-be 4 strings? or maybe with multiple singers? does switching on more than one mic at time ever create weirdness? I mainly ask in regard to possibly having wiring/polarity/phase issues with maybe poorly wired XLR cables, or speaker cabs not all playing well together in opposing polarity/phase +/- cone action. Could be all sorts of things out of wack like that. Some church audio systems I've seen are very piecemeal - installed in bits and pieces over time, as well as with gear ranging from retro archaic to modern high end - sometimes all mixed together. Making sure all of that is in harmony might be something to consider.]
Nevertheless... The first thing I'd want to do is look to the DI box. Does it have polarity or phase switches. The signal from the bass rig that the player is hearing - maybe not so directly with where they sit/stand in relation to their amp, might be not playing at it's best with the DI box, having problems with how it translates through the PA, if some speakers aren't playing well together.
idk, just throwing out something that seems less mentioned in the thread. The DI and how it plays with the system. The flip of a switch, if it has one or two or three, with each different bass - might make a difference. Is it active or passive DI. Phantom power or depending on a diminishing battery? Is the cabling for the DI ok? XLR with grounding issues? Swap cables. Swap DIs.
Something you did mention, about room acoustics can be part of the problem, are there room issues particularly unfriendly to bass freqs? Lower overall volume a tad.
-- I had to do some theatre audio, where the actors were all on wireless lapel mics. I spent hours trying to get the audio evened out with system and board EQs due to mic movement - but I couldn't kill this particular resonance. All the mics were omni directional, it was a real bear. After working late to get things ready the night before show, I shut things down, figuring to come in with fresh ears. - Everything off, and as I was locking up, I banged a table or chair or something and BOING there it was. It was the room - it had a little extra rim of drywall above the stage area, maybe for covering conduit, it was a freq trap - no way to kill it without acoustic treatments. All I could do was live with it and keep the volume mellow. It was a small room so it all went well.
Some different ideas to consider. I don't know for single bass string harmonics canceling each other out. I can see it on maybe old string(s) on one bass, but across the board, i dunno... If your problem is all basses, follow the trail from amp/DI to PA. If your problem might effect other instruments or mics, scrutinize the PA wiring or speakers and their placement. Lots of possible gremlins. I don't think it's the basses. Now the bass players? Maybe.