I don’t have any experience of this I’m afraid, however I am on the Board of a small care home.
I’d advise you to establish who your customer is. You might prefer to just ask the staff at the Home what is required, and get them to settle up, but this is not respectful to family wishes and concerns. It may be that some of the residents don’t have enough independence of mind to decide what they want and their family may be the ones making all the decisions. In this situation, you cannot contract your services with the resident, or the home, it’s unethical. The family may be the ones specifying the service and you may have to negotiate with them if you think that’s going to be tricky - for instance some services may be too long for disoriented residents. It sounds disrespectful, but it’s a bit like having a dog grooming parlour, you take the wishes of the paying customer into account but you always put the welfare of the one receiving the service to the fore. If another party got involved with the service you’d feel anxious that there’d be a miscommunication somewhere along the line.
I suggest that you set up agreements direct with the families, agree what services they wish for their loved one and yes, send them an invoice. It needn’t be onerous. You send out a monthly invoice to each family, with reminders. Make it easy to pay - give your bank details and a reference number so they can pay by bank transfer. You can be clear that you won’t do services on residents that haven’t had outstanding bills settled. Maybe look into a direct debit or standing order arrangement.
It’s actually a fantastic opportunity to have a regular income - if you’re sick, you just reschedule your day direct with the home, you won’t lose any business.
You could ask the home what would happen about unsettled invoices and see if they have a resident fund that could be used if you feel very anxious. Bear in mind that if they pay you for your services, and agree a regular day for your visit, and start specifying what you do for a resident, they may potentially be creating an employer employee relationship.
Lots of very ordinary services work on an invoice basis, my window cleaner and lawn mowing chap leave me invoices rather than expect cash in an envelope. I expect the Home has a similar arrangement with visiting chiropodists other allied health professionals. It’s their normal way of doing business.