Yes, I'd go along with what boxtops is saying.
It sounds like we have a charity HR department that doesn't know too much about employment law - what do they think a 'rolling contract' is? - every employment contract is a 'rolling contract' in that it continues until terminated by one of the two parties!
Flexibility is a great phrase with no clear definition about its scope. I think you would find an Employment Tribunal (ET) would interpret it as the employee being willing to deliver the contract in a flexible way - so temp changes in start/finish times, or days on/off, and maybe a SHORT term acceptance of additional hours - but not to an increase of 100% for a long period.
The trouble nowadays is that, with jobs in short supply in some parts of our great land, people feel their job could be at risk if they don't comply with what the employer wants.
If your manager was to leave and you think you'd be asked to 'fill-in', it seems clear to me that this is a different job - different job title and different pay rate?
The way I'd handle is be upfront and pro-active and approach them. Outline the situation, your keeness to help out on a very short term basis, that you'd be willing to consider an upgrade the shop manager role, subject to discussion on a change in the pay rate and an understanding of the required hours. Stress your commitment but also stress your needs regarding a family life. State you would be willing to agree to a maximum of X weeks at it.
Keep your own notes of this conversation. You'd be happy to consider a temporary change in contract on this basis, subject to you seeing their letter outlining these changes.
By dealing with it this way, you separate the two jobs, and it requires them to put this in writing back to you. That should limite the opportunity for scope creep.
You have been working there continuously for more than 12 months, I trust?