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You can turn mobile access off in the car, but I think that disables using the app for a lot of things. (And the OP didn't want it to be obvious that they were blocking the location information.)
what all does it disable when you turn off/on mobile access? can't see where it's a big deal to turn it off and then back on if just occasionally wanting to be stealth.
why do your kids have acces to the app ?
If you don't want them to see where you're at and still want them to be able to drive the car, just give them the card when they need it