Firstly, there's not really much point your MIL driving to the hospital the moment you're in labour. You'll probably labour at home first until you're further along and then once you're in hospital they don't like people just hanging around. So unless she's your second birth partner, she's probably going to need to just sit in the restaurant (boring!) and labour can take a while. Your DH will be in the room with you, supporting you, he won't be out in the restaurant getting support from his mum. The only time I could think of would be if something untoward happened, in which case he could call her and she could come to the hospital. This is unlikely though. If your DH is that worried about labour, then I'd definitely suggest he watches some home births on YouTube, there's lots and they're often very relaxed.
The midwives are usually very good at giving you some space after the birth anyway, so they're unlikely to let DMIL in straight away, you should check their visiting policy, then no one is disappointed.
I'd actually suggest that your DMIL picks up your DM once you've had the baby and they both come together for visiting. Do you really need your MIL to drive you home after? (getting discharged is often a pain and either takes ages or you're heading off with zero notice!). Alternatively, if DMIL does have to drive you then politely ask DH to ask her to drop and go so you can have some family time. You can then make arrangements for both DM and MIL to visit. I agree if you need emergency childcare and/or have a second child, your MIL will be an additional resource to help with that. Don't cut off your nose to spite your face.
TBH once you've had the baby, so many people will have seen him it probably won't matter who sees the baby first from the family. DD was held by midwives, paediatrician and probably others before family. Also, I'd had my norks out to everyone too trying to establish breastfeeding, that I no longer cared who saw what by that stage! Often you feel different after the birth, you'll be pretty knackered by then too, that you might find you just want it to be you and your baby for a while, until you've had a little sleep, some food and possibly a shower! I'd leave all the minute visiting details vague until after you've had the baby and know how you feel at the time.