Stand your ground on what YOU want for YOUR Christmas this year. Seriously. If your partner wants to continue pleasing his mother over his wife then let him, he can go to mummy's for Christmas and maybe then he will see just how unreasonable and unfair she is being. She will spend the whole day complaining about you and how you've stopped her seeing her grandchild - hopefully he will stand up for you and back you up on your reasons, but if he comes home and has a go at YOU for staying home, warm, and in your own familiar environment with freedom to do what you want, with a 5 month old baby for your first Christmas as a parent, then I'm sorry but you need to rethink the relationship. He left home and got married. There should be a new woman as priority in his life, and it ain't his mum.
You need to learn to drive too, and really drive it into him that he gets rid of that bike. There's no way he can't find time to let you get driving lessons, or get them himself, unless he's working a 90 hour week?! I work shifts and there's always time for important things like that. You have a child, a car isn't a necessity but it sure as hell helps when you want to go out anywhere. When that baby gets a bit older and you want to go to the zoo or for a day out to a park, you're going to drag a pram and all your baby stuff and possibly a picnic onto a bus or into a taxi? If you can afford a car then trust me, it makes it 10 times easier.
Those saying you're unreasonable have either never had kids or are just as selfish as your MIL. Who invites someone over, offers to drive them there and back, and then drinks so they can't take them - a BABY - home? A selfish, inconsiderate cow, that's who. You don't need that in your life, and if that's how it's been for only 2 or 3 months, it's going to get a lot worse the older that child gets. Trust me, I know what I'm talking about when it comes to selfish, entitled, self centred MILs!