Firstly, it's really hard to judge this accurately, when you haven't told us why you are sleeping apart. If it is something that means your relationship is damaged beyond all repair, then I think you should split up. That doesn't necessarily mean that either of your moves out - it could mean that you lead separate lives within the same house for a period of time. You could get a solicitor to work out the financial aspects, but if you had completely separate financial affairs, then he would need to get a job or claim JSA. You would probably have to give him some financial support - what would you have to pay someone to look after the dc while you are at work? But otherwise, he wouldn't be your responsibility. That could only work if you both agree to it, for the sake of the children's stability. You would feel less resentment, because you would not be financing his holidays etc.
I think it's the separate rooms issue which is having an impact on everything else - I think you are basically feeling that you are paying lots of money for him to loaf around and you are not even in a proper relationship any more.
In principle I cannot see a problem with one person lacking ambition in a career sense, if their partner has it, loves their job and earns enough money to financially support the family. That said, as a sahp he ought to be doing the bulk of the house work and child care. I honestly cannot see the point in him getting any old job, just for the sake of it, when money isn't needed.
You need to think about this really carefully and with a solicitor, go through all the possible outcomes, before you act.
I would start with relationship counselling - you say he is a good dad and not a total waster, so for the dc, you should try and get the nicest outcome, if possible.