OK. You can do this, but it will take time. I used to drive to DM - 260 miles - every 6 weeks or so with DS from about 5 weeks old.,How are you feeling post birth? Are you a confident driver? Is there anyone (adult) who coukd come with you?
Could you go by train?
Could you stop in a Premier Inn type place overnight and fo two shorter journeys?
Do you have friends or family on the way who you could stop with for a coffee/overnight?
A 5 week old should be fairly straightforward - at least he won’t want to be out there and running around.
Plan your route. Even if you know it, I used to sat nav it anyway1 - to remind me to leave the motorway if I was distracted! 2- in case I had to change route for closures etc.
Plan where you are going to stop to feed/change nappies/wiggle around. Don’t forget to feed yourself. Have other “backup” stops in mind in case you or your baby needs it. When my dc were small, advice was not to spend more than 2 hours in a car seat at a time. We used to stop for 20-30 mins every 2 hours. I don’t know what current guidance is, but I’ve seen you can get lie flat car seats now.
Download something interesting to listen too or chose some music you enjoy.,
I used to take DS swimming beforehand to tire him out, then travel early evening whilst he slept. He was very easy to transfer to a cotbince we arrived though. 5 weeks might be a bit young for swimming (although DS went at 6 weeks after his first vaccinations).
That’s my practical advice. I know that won’t help emotionally, and that every baby is different. I hope it goes well. Good luck.