HB is based on income NOT the landlord or the tenant being a Lp.
He loves his kids so much yet he has to be MADE to see them? A lot of nrp's would love to see their DC every weekend.
The debt was his choice/responsibility as are the DC who need a roof, home, furniture and deserve to have their lives disrupted as little as possible due to their parents splitting up.
Entirely possible the exgf can't afford the transport. Does she work? Are there time commitments re her doing drop off/pick up? Or one of them? Legally all she has to do is make the DC available.
Re working overseas I agree with exgf. Terrible timing for the children and while nrp's should pay a decent amount of maintenance, money isn't everything. This would be a huge wrench for the DC.
You say he COULD get a flight back every 2/3 weeks? But WOULD he?
4x more pay sounds great BUT he'd have to pay rent and bills for a home out there (and even if he's paying you keep it will be much more than he's currently paying. Plus places with high pay tend to have high housing costs). Rightfully he should pay more maintenance too, and he'll need to pay for flights back, and pay more toward the debts... So that "extra" money will soon disappear, he could even be worse off!
He should see a debt counsellor, look for better paid work nearer to you, appreciate the time he gets with his children.
She's NOT being grabby. (So much for neutral eh op?)
Frankly it sounds like a barely concealed excuse for running away.