Here's some advice on how I position MY drums:
1. I put MY drums in the car.
2. I take MY drums to the gig.
3. I set MY drums up for the gig.
4. I play MY drums on the gig.
5. I break MY drums down after the gig.
6. I take MY drums home.

But seriously. My main kits are fusion kits. I hate playing on standard size toms (12/13) because they're so deep. My 1st personal set, and our main church set are standard kits. But I added a 10" tom, and cut the 12" tom from 10" depth to 8" depth. But I'm the drummer and the pastor trusted me to get it done professionally. I HATE standard size toms.
When I'm faced with having to play someone else's 12/13/16 kit, I usually don't even play the rack toms. I completely ignore them and use the snare, hats, kick, & floor tom. This may take some getting used to. What I did was break down my practice kit to nothing but snare, kick, hats, and 1 crash. You'd be surprised at how this limited set boosts your creativity. There are fewer toys to play with, so you're forced to think outside the box, so that your playing doesn't sound redundant.
My church is visiting someone else's church this Sunday, and our choir is singing. Whenever we visit this particular church, the guy's drums (12/13/16 toms) are always tuned like marching band quads: really tight, and high-pitched. I can't stand that sound, and won't subject myself to having to hear it while I'm playing. So I just play kick, snare, hats, and crash. We only sing 1 song so it's not worth taking my own set. But I still feel comfortable with my snare, so I can groove like I need to.
If you're playing more than 5 songs at these gigs, I would suggest taking your own set. Or, if possible, taking your own toms (I've done that before, too). But that requires that the other set has the same type of tom mounts as your set.
These are just some suggestions. Hopefully someone else will chime in and help you with your dilemma.