Why should you be the 'weekend' carer? Ds1 is unlikely to see much of his mates after school on weekdays as they'll all have homework, ds2 won't be able to spend much time in the park on weekdays as the nights are drawing in, and dd will be nearer to her df for longer when he's not working which, presumably, is at weekends.
If you point this out to them they may start clamouring to spend long weekends with their df - he could pick them up from school on Friday, take them to school on Monday morning, and you have them Mon thru Thurs.
While your ex may be rubbing his hands with glee at the moment, I suspect that a couple of weeks of him having them every weekend will soon wipe the smile off his face and, more especially, when he realises that you're free to go out and 'mingle' while he's a sahd.
Does the 'shit tip' have the potential for improvement? Can you involve the dc in choosing paint/soft furnishings for their rooms, get them enthusiatic about redecorating, and generally work towards making it a home you can all be proud of? The thing about crappy areas is that house prices can only go up they can become gentrified and painting the front door in a Farrow and Ball shade and sticking a couple of tasteful planters either side of it can start a trend even if you do have to chain them to the wall, or cement them in place, to stop them being stolen. 
That said, I suggest you tell ds1 he'll be free to decide who he wants to live with when he becomes 16 and tell the other 2 they can't always have what they want. What would be patently unfair to his younger siblings is to miss out on having ds1 as their big bro role model and maybe you can use that persuade him to give up any thoughts he may have of moving out in the near future?
In law ds1's wishes would be taken into account, but that's not to say that a court would take the view that he should live with his df and, from what you've said, it seems unlikely that your ex would want to have the dc full time.
Of course it's hurtful for you to think the dc don't want to be with you because their df's house is 'nicer' but, as independentfriend has said, it's more about them wanting to hang on to the past and what's 'known' than materialism on their part and I have no doubt that none of them actively want to leave their dm - they're children and they don't yet have the depth of experience to think it through, which is why you have to do it for them and put your foot down if necessary.