Have you looked into discretionary housing payments? If not the contact details should be on your letters, or the council website etc.
It's usually a one off payment for a fixed shortish term, and it's meant to cover situations like yours, or eg someone losing their job and therefore having a gap between lha and rent. They are usually only short term, and how generous they are varies between areas and how much they have in the pot.
In your case you have the reason, and of course even if your dd didn't have health problems, it wouldn't be practical, cost effective or likely possible to move to a 2 bed till they are 10, then attempt to find a 3 bed in budget that takes hb. So you do meet that criteria.
The means criteria are also quite tough, you need to provide a breakdown of outgoings, and they have to agree you can't cut it back to cover part/all of the shortfall. If you are at all unsure do get help from the cab. In some areas if you write £25 for food for 3, and £5 for both kids at a club, they'll tell you to ditch the club, even though they'd accept £40 for food. And if you do it alone, include any evidence for anything they might question to save delay. Eg a broadband contract lasting another 6months etc, loan repayments, otherwise they'll write back saying to stop those payments and put it towards rent.