It's a difficult one, without knowing the full circumstances.
The cause of his current condition is really pivotal, TBH. As is the type of ward he is in now. I assume he's in an acute geriatric/medical ward.
He could be presenting in this way because he has an infection or other illness- but it could also be delirium. One of the (many) frustrating things about delirium is that it can persist for quite some time after the initial trigger (e.g. an infection) has been treated and has resolved. It can also be intermittent, or re-triggered by really minor things (constipation, really mild viral infection and so on) after the initial episode has resolved. If your dad does have delirium triggered by the infection that has resolved, and is otherwise not acutely unwell, then it could be that a nursing home (or rehab bed either in a rehab ward or in the local community hospital if you have either locally) may be the better option when compared to a bed in an acute hospital ward. This is because the acute ward is likely to be busy, noisy, with coming and going all night etc and this kind of thing can worsen/sustain delirium. The things that help, settled environment, low lighting, lots of reassurance and explanation, are far more difficult to do in the busy acute ward with lots of new patients coming in/patients being very unwell and so on. For patients who have persistent delirium, it can be more appropriate to look into a quieter setting. Whilst in an acute bed the risk of hospital acquired infection is also present, so this needs to be considered too.
Shingles can be treated with antivirals if caught early enough but not much else can be done (assuming he has not developed any complications such as neuralgia) so again this in itself should not prevent discharge to appropriate care setting in an otherwise physically well patient.
I can understand why not being able to mobilise independently would cause concern- but again a rehab bed would be more appropriate for this problem than an acute bed. In a rehab bed he could have physio/OT assessment as well as nursing care and community rehab teams can go into nursing homes to provide physio etc there.
I think you should ask for a meeting with your fathers team and find out why they have come to this conclusion. For example, I'd want to understand why they are suggesting a care home- is it because they think this is delirium which is going to take time to settle but there is nothing acute (in medical terms) which requires in-patient treatment? If that is the case, what are the plans regarding rehab? What rehab resources are available in the community? Or do they not feel that he has rehab potential? If not, why have they come to that decision? It would have been better if they had explained this already, of course.
It is, of course, possible that he is being discharged prematurely due to bed pressures or something of that ilk, but I wouldn't jump to that conclusion immediately. Speak to his team and go from there.