
As said, there's a world of difference between a puppy being left with his siblings in familiar surroundings etc. and being left alone without his siblings and in totally unfamiliar surroundings.
Yes use a crate BUT you can't really expect a puppy to accept finding herself in there, again on her own in a strange place. With our buy-ins (when we had our home-bred puppies we almost always kept back 2 and they were in our puppy pen), we set up their crate in the living area, with top and sides covered and the door open so they could use the crate like any other dog bed, going in there to nap. We didn't go out together, leaving the puppy alone (even if he/she had our other hound for company) for some weeks at first, so when we did finally go out together, after playing with our baby Whippet (9+ weeks) and letting her out to empty, she went into her crate with a few treats and the door was shut. She didn't much like that, but we went out and were gone for about 1 hour. We came back expecting to hear a riot going on - not so .... she was fast asleep and didn't even hear us come in. We left the radio on, and she had old then old Basset in the same room with her.
What's going on with this poohing would most probably be her form of separation anxiety. I've known an upset youngster 'wring out' the last drop in protest!! And that should be avoided at all costs because of all places, her crate should be the one secure safe place for a puppy to be.
I'd certainly not leave her alone in the car, crated or not. We had to do this one time when, with only 2, one took off so we put the other in the car so we could go back to find him. When we got back he'd chewed the brake handle, the steering wheel, etc.etc. And he wasn't 10 weeks - he was a boy we'd had to leave back with his breeder after he was too sick to travel to our new home abroad with us. It was his form of protest.
Fact is you may have to clear your appointments for now and concentrate on getting her settled with you and into a good routine. If you can't, can you have her breeder take her for the time you'll be out? Just this one time?
ps I appreciate your concern re renting!!