is Bahrain friendly to working moms?

Yours is a very hard question to answer without knowing who your employer will be, what sort of job you are going to do and what your contract says about hours, holidays etc.  As in the West every employer is different.

Different schools start at different times - my son starts at 8am - and it depends how far from school you live as to travel times.

Daycare here is limited.  But most families employ full time maids to look after their children, do the housework and sometimes do the shopping and cook.  If you employ such a maid I guess you will have more time with your family even if you work full time.

Scottrace gave you a good answer - it really does depend on how flexible your employer is. My kids are older so I haven't experienced day care here but most working mums do get a maid/nanny and also a driver if the kids need to go places while mum is at work.

Overall you would definitely get to spend more quality time with the kids - your schedule sounds really tough, I'm getting tired just reading about it.