You said in your OP that it's a provisional offer. I would find out what would happen next - if you have to go through security checks etc, that can take many weeks, running into months. You are advised to not hand notice in at your current job until that process is completed, presumably because a) you might not get cleared! and b) because the checks can take longer than your notice. So find that out first. You can go through this process without letting your current employer know, and make a decision later.
Secondly, as PP has mentioned, £28k now may rise to £30k fairly soon after you start the role, after going through security clearance and then notice after that. There may be scope to advance - either in salary increments or by promotion opportunities. On top of that, the pension is decent and there are lots of other benefits.
Once you are in CS, there is a lot of scope for applying for flexible working. This is separate from flexi-time - I think most if not all depts allow that, and as PP explained, you accrue flexi-time by working a bit longer (as the business needs, not in order to build up a day off!), and then you can apply to take a flexi day, basically same as TOIL (time off in lieu).
Flexible working is different - you can negotiate this after receiving your firm job offer, or if that's not agreed, after you've been in post for a while and can justify how it would work for the business. You could, for example, request to reduce your hours (and pay) by 20%, which would be £22.4k, almost the same as what you're on now (but with better prospects and pension) and then work 28hrs over five days while your DD is in school, minimising need for childcare. Or you could request term-time working, or parental leave for school holidays... there are a lot of options and if they like you, they are likely to be open to discussing what works. Many of my colleagues have a flexible working pattern - I do compressed over four (long) days, so am on a full time salary but have a 3-day weekend every week. I have friends in the private sector who earn more but also work a lot more hours than I do, so their 'hourly pay' isn't much better.
All of that is to say, don't make a snap decision. I think you should find out what would happen next, and take a longer-term view. Your current role isn't likely to start paying anywhere near what you could earn in CS within the next few years, especially if you can move up the grades, and the additional benefits are worth balancing. Imagine being in the position where you can take your DD on a great holiday in a few years because you've got enough disposable cash, instead of just being off all school holidays but unable to afford to do much. Or maybe she'll want to start an activity and you can afford it, instead of having to say no or cut corners. Short-term pain, long-term gain. Good luck whatever you decide.