I chose option 2 when in your shoes. What I would say is be prepared for things to change. I had a clear agreement in place with my mum before moving and then it quickly became apparent that the reality was not what she'd imagined and I then had to pay for full time wrap around childcare anyway which is expensive.
Equally, if I need the odd bit of childcare I now have options for that so I can go to the hairdressers, work events, the odd social thing. Which gives me more balance to my life. This is so, so important when you're a lone parent.
7-7 2 days a week is a lot for you but I'd say you are better not setting yourself up to only ever be an employee or a mum, which is what you'd be if you are city based with no supports. You need time to just be yourself as well. So being closer to the supports that enable you to do that is worth its weight in gold. Even if friends nearby can call in for a cuppa when your child is in bed, that's a very important thing to have in life.
So I'd say you need to ask your mum to sit down and have a very realistic think about what she is actually able/ willing to do in terms of childcare so you have a clear understanding of what that will actually look like before you move, cost local nurseries/wraparound and make decisions based on that.
The fact you have friends nearby also means that if your mum is busy a day you might have a back up. I'd be also offering to babysit their kids as soon as you're settled to earn a few favours.
I'd speak to your work and see if you can spread your hours.
It will also be worth speaking to a citizens advice service to check what benefits if any you're entitled to. I got much more help than I'd expected and it's been a godsend.
It's a tough adjustment and initially I found the first year I was really just surviving and living on routine, but a few years in and I'm focusing much more on myself and life feels good again. Time is a great healer.