To me, this is a complete overreaction.
He's a great dad, great around the house.
He has an achilles heel, which is that when cornered over relatively trivial things, he lies.
He is not lying over affairs/losing his job/big troubling lies.
I don't count knocking £40 off the price of something as a serious lie, I do that all the time and I know my husband does it to me, we mean it was 'about' £40! It's a white lie and I don't think this is a hanging offence.
With the car, I obviously am a very dangerous person with no regard for my childrens' safety, as I have started leaving my 6 year old in the car when I pop my other child into ballet. It takes about five min, she's very tired, and I have made the judgement call that it is safer for her to stay in the car with her book than get out, cross a busy road twice and come with me. It is 5 min max and that's what I feel is acceptable. My eight year old often reads her book in the car in the same situation. If a policeman came along, they would see a child sitting in the car, reading a book, with the window open, and I would be back within a few minutes. So what?
Now- he shouldn't lie about it. And, if he's a good parent, and the other parent expresses real concern about this judgement call, he should do what you want, out of respect for you (my husband on the other hand thinks we are absolutely bonkers about safety, as he comes from a European country in which 6/7 year olds walk to school, all of them, go away from home for days with the school, and go on skiing trips whilst in nursery, so has an entirely different understanding of what 6 year olds are capable of).
Sit him down again. Say: you are really pissing me off with the lying. In particular, you know I don't like you leaving X in the car, so DON'T DO IT. And, I will be checking.
The other lies, be honest with him about how much it upsets you.
But really, do people honestly get up and walk out of perfectly ok marriages (which could be worked on) for stuff like this? Will the children really be safer/protected from this (nice) daddy who happens to tell the odd porkie if they are separated, with all the trauma and distress that brings even if handled well? And, let's face it, if you separated, he has sole care for his portion of time, he can do what he likes and say what he likes!