It's just a crush that will be shortlived, honey, although of course I am adorable and immensely loveable 
You mentioned 'bedrooms' and he's a single person? It's belatedly occurred to me that your local authority will have limits on the amount of HB payable for a 1-bed property which is all he's entitled to and any shortfall may be far greater than what he may be anticipating.
I suggest you give your council's HB department a call to find out what the maximum sums are for 1-bed properties and however many bedrooms the house next door but one to you has.
It also occurs to me that he may try to screw the council claim he needs 2/3 bedrooms to accomodate his dc. If this is the case, you may need to make another call or ask your solicitor to write a letter to the council making it clear that, as any contact he has with his dc is supervised and does not include overnight stays, he doesn't meet the criteria for entitlement to more than a 1-bed property.
I get the feeling that maybe the landlord is having second thoughts about having your ex as a tenant which is why he's made excuses not to proceed immediately, and I don't think it will take much for you to put the boot in ensure that a more worthy tenant becomes one of your near neighbours.
Some twunts have an uncanny ability to rear their ugly heads just when life is looking up for us. Tell your twunt that moving so close to the dc's home would be deeply unsettling for them and that there is not a snowball in hell's chance that such close proximity will in any way lead to him being included in your lives any more than he already is.
Also make it clear to him in no uncertain terms that you've had ample time to reflect on what a mug you've been in the past and that, as he no longer has the ability to play you like a violin, he'll have to find another instrument to practice on.
For good measure, add that he will not be getting one penny of the lump sum you are expecting, nor will you subsidise his idle lifestyle in any way, shape, or form at any time.
Don't despair, sweetie, and don't torture yourself with thoughts of having to move because that ain't gonna happen until YOU want it to.