My first reaction is that your FIL is not the boss of your MIL and telling her what she should/shouldn't do is not on! Surely she can make up her own mind?
Personally, I wouldn't dream of accepting money for babysitting my GC but as you don't mention your IL's financial situation, it may be that they need that extra income, however small, so that's a factor I'd consider.
I think I'd approach it by saying (or your Dh could say it, as they are his parents!) that finances are going to be tight when the new baby arrives and you won't be able to afford to pay for babysitting, or only a reduced amount. I certainly wouldn't argue about it. I can't see that anyone would be a winner if that were to happen.
Then I'd also take a look at the date night aspect anyway. Are there other things you could do instead of going out? Setting aside a regular evening to have a take away meal, sharing some wine and watching a nice film at home could be an alternative arrangement. I know it's not the same as physically going out but it's a compromise and anyway, I don't suppose you'll want to go out that much with a brand new baby - I remember always been exhausted by the evening and ready for bed myself! Babies grow up pretty quickly too, and you'll soon be able to go out again and enjoy yourself.
Another thought is that maybe you could join a baby sitting circle, although I have no idea if they are still a 'thing'. That was my main source of babysitters when my children are small plus it was free.
I do hope other Gransnetters come along with some alternative ideas. Good luck!