Unfortunately you don't really have many 'rights' here, if it is in your contract that your employer can require you to work bank holidays which it will be, then you are a bit stuck if you refuse and will likely be a serious disciplinary or even dismissal if you don't turn up for your shift or call in sick etc. as it will mean residents could suffer and colleagues will have to cover your work etc. which really isn't fair on them. You have a statutory right to emergency unpaid carers leave for sick DC or childcare failure but really this is not an emergency as you know about it in advance....
I think your options are in order of preference
(a) Go to your manager and beg them for the time off, offer to work the next 2 Christmases (your DH will need to arrange time off or childcare, he will have a whole year to do it), all bank holidays in 2020, whatever it takes. Offer to work a few hours on the day so you can sync up your and DH's shifts e.g. you work the morning he works the afternoon/evening?
(b) Your DH does the same, he surely can't be the only person in the whole home who can do medication, what happens on his days off?
(c) Beg/persuade your colleagues to swap with you for at least a few hours of the shift, offer to work whatever graveyard shifts they hate for however long it takes etc? May make you unpopular as obviously it puts them in an awkward situation but needs must if you have no alternative.
(d) Find childcare for the day - what happens to the DC when you are both at work normally - any grandparents/siblings/friends who could have them all or part of the day -split up if needed? I know it sucks but needs must to stay in employment?