I'm a pimp who used to be a programmer and although I don't know your boss, I'm very sceptical of promises made to retain IT people. There is also a commonly held figure that about 2/3 of people who are draqgged bakc like you are gone within a year anyway.
None of the firms we deal with ask for medical reports for anyone earning less than 75K and that's an insurance requirement because of the large exposure.
However one firm we never ever deal with is EDS, and they do this sort of thing.
I don't know why I keep butting heads with Senapostrophe, but she is dead wrong on "career suicide", the game is very buzzword driven, you need new ones all the time.
Frequently I come across people who've slipped behind and it can be quite sad. And yes it happened to me once, was quite bloody.
It is very hard to get good skill improvement in your "spare time". I found that when I was single, for someone with children, it's almost certainly just a route to stress and exhaustion.
Employers give little weight to self taught skills, and usually demand that you've done it in "real life".
It partly comes down to how much you trust your boss, and whether he actually can deliver on these promises, not just if he is honourable.
You also need to get some observables in this process. A specific commitment to training, and provision of equipment for home working. And for the avoidance of doubt "training" is not a vague promise that if something suitable turns up they will try. It is you finding a course that extends your skill base, and chose by you for that specific purpose.
You need to do some calculations on child care, and include in that cover for sickness etc.
Thus you should not turn down the big firm yet.
If you want a chat in a less public forum, I'm [email protected]