This is a topic close to my heart as we have been hit significantly with huge travel costs for well over a decade as DP's ex moved several miles away (without compelling reasons like work or housing affordability) and then completely reneged on her promise to do at least half the driving. The cold hard fact is that if we don't pay DP doesn't get to see his kids but on the other hand as we are counting every last £ this expense has also meant my children (one of whom is also DP's) inevitably miss out due to his ex's selfishness. We are easily talking 5 figures here in travel costs - and still rising. It's a huge sum and I readily admit I resent it very much as it was none of our doing.
I do personally feel that unless it's completely avoidable the parent doing the moving should bear a greater moral responsibility for facilitating contact - I accept people want to move for all sorts of reasons, not just when they have no choice - but believe that when you have young kids the impact upon them needs to be considered and minimised as far as possible. To be fair, I do recognise that you discussed this with your DP's ex beforehand and you weren't to know that the agreement she made would later be broken. It sounds to me as if quite possibly she "agreed" on her boyfriend's behalf but hadn't actually spoken to him and/or he hadn't actually realised how long the journey would be. I also recognise that you've upped the maintenance to cover off the ex's "half" of the travel (which is more than DP's ex has done, i.e. there's no way in the world she'd ever voluntarily agree to a reduction in maintenance to cover our costs) but agree with the other posters that this really isn't the ex's BF's issue to sort out (regardless of his own contact arrangements for his child).
Have you asked if ex would consider learning to drive ? A big ask, but she may do as this would benefit her in all sorts of other ways and perhaps you could make a contribution to this ? A bit annoying but in the long run if she passed her test and did half the driving it would help you lots.
The other thing as others have said is whether given the distance it's cost and time effective for the kids to be brought back to yours at all ? If you took the "extra" maintenance and DP used that to book a Travelodge for a couple of nights, maybe he could see the kids that way and he'd get a break from driving. i.e. 4 hours up - 2 days rest - then 4 hours back as opposed to 8-9 hours all at once (plus loo/food stops).
How old are the kids ?If older, could they travel by coach to a point closer to yours so you could collect them from there ? (suggest coach instead of train as I feel it's safer, they're less likely to get off at wrong stop etc).
Hope you sort something out - obviously don't know the circumstances of your move and though it's tempting to say in effect you've brought this upon yourself, I don't think it's entirely fair to do so as when you did move you took ex's agreement at face value - and at the end of the day, it's the children who are missing out on a relationship with their dad.