Your second paragraph is not clear so can't advise you on that, sorry!
WRT flexible working I'd apply officially rather than doing it off the record; if a flexible working request is granted its a permanent change to your contract so future managers couldn't make you go back to 5 days a week, whereas if you are doing it off the record you are potentially setting yourself up for gross misconduct accusations! If HR say no they can't force you to go part time.
Have said I will go back to work beg of march and have asked to do 5 days in 4 as I work from home full time so no travel time. Job v sympathetic to having children too but head of dept and supporting head do not want me to do this via HR in case they made me go part time which I can't afford with 3 kids. Do you think this will work if not officially done through HR? They've asked me to not tell the rest of the dept and to put myself in meetings all day on Wednesday and to be creative with my time sheet on that day (do consultancy for clients).
Before I went off on mat leave, the head do dept 'left' and dept was restructured. Half of the staff are leaving due to this new change and I am concerned my new manager is going to be this woman who I don't like ( was consulted on who lid like to manage me, supporting head do dept said shed see what she can do ur no promises. They don't want this woman being without staff and supporting head said I could stand up to her but I said frankly that is exhausting!! Anyway, am being contacted on linked in by CEO of another company asking if I want to catch up and they had a no of roles in their company that may be of interest which I know would pay more money and would be a promotion. I have really enjoyed where I am and they have been good to me - but with 2 out of the 3 people gone who attracted me to the company in the first place and with impending changes, really not sure whether I should simply sit tight and see how it pans out or whether I should explore these other opportunities? The new company has less of a good rep, but the CEO used to work with with me in a previous job and has been given 100% approval on glassdoor and I wonder whether the company is on the up? No doubt the job would mean some office working so bear in mind I have 3 children under 5! Any thoughts?