Flexible working request - reasonable?

Hi everyone! I am based in the UK and due to go back to work in November this year. I would like to go back part time (I mean, ideally not at all, but bills need to be paid!) and am considering my options.

Key facts:
- my workplace is office based, no working from home at all. They will not budge on this
- the commute is around 1.5hours each way from our new home
- full time working hours are condensed partly with an early finish on a Friday, making 37.5 hours total

Due to the long commute, it starts to become a bit ridiculous thinking about going back full time. I am lucky in that my partner is self employed and so, with planning, will be able to look after our daughter when I go to work.

So, I am considering dropping to 16 hours per week, which would be 2 full days Monday - Thursday. It wouldn’t make much difference to me which days they were so the business could choose which was best, but does 2 days sound unreasonable? Other people have dropped to 3 days but I would be away from the house from the time my daughter woke up until bedtime and this just sounds horrendous to do for more days. I feel 2 days wouldn’t make me miserable and would be financially worth it.

My alternative is to find another job, my industry is not too bad for this but having said that most places are looking for full time rather than part time.

Any advice/thoughts welcome. Thank you :)