Cat... Here is my advice and I am wishing for your best life:
please get a hold of your financials. This is good for any woman to know anyway. Make sure you can get whatever documents that show your accounts or whatever.
Women should all do this, it is important even in the best of situations.
I wouldn’t have it out with the kids in the pram or whatever. You know deep down this behavior isn’t normal.
Get all your ducks in a row. He will keep lying and praying upon your trust and understanding. The less he knows of your suspicions the better. It’ll feel good in the moment but it is SO MUCH BETTER to really have the conviction and knowledge when you do have a talk.
It’s not normal he should ride with anyone and not know their last name. It’s not normal someone giving a ride should not go 5 mins further by car to pick that person up at their house. (Odd to do kindness for someone and not drive to save them a long walk)
It’s odd that you both wouldn’t pay kindness back to this carpooler. Why has he not wanted to buy this man anything or have you or him bake anything? Like someone is driving him round saving money... wouldn’t it be nice to invite him round for dinner or whatever? That’s weird. You know it is.
If you haven’t tipped your hand or made a scene you are in a very powerful position. I know it hurts. I KNOW it hurts. But if I could go back and give my own self this advice I’d have cut out a lot of BS.
Once you have some ducks in a row. Quietly unfriend her without comment. You already know she’s weird about it anyway. Watch as he skitters and wonders.
When you have everything in a row and a good plan then go ahead and see the horrible crap for yourself. Go and see him hop into her car. But at least if your ducks are in a row you won’t be left rudderless.
Also (sorry so long and again this is just my advice but I feel for you) please don’t have unprotected sex with him. Cheaters cheat and they cheat even on the ones they cheat with. And the ones they cheat with keep cheating as well. Don’t put your body in danger.
You’re worth more than lies and gaslighting. You’re worth more than having messy fights that lead to more lies. You need to have yourself a good plan and then you strike. If you don’t you will ardently want to believe lies. If it’s genuinely a mistake then you will have the safety of your plan to let you do things rationally.