Sorry to be so tardy replying...
As I mentioned in the first post, I do know this sounds daft - but desperate times spark desperate measures. As I've been unable to find the person I need in Switzerland, I started thinking about whether it would be feasible to bring someone in from elsewhere.
Not wanting to veer off the topic of how to do so, but to answer your questions...
Yes, I have dogs with medical conditions or who still carry some of the baggage of their past lives; a Tierheim is just not equipped to provide the level of care and individual attention they need. In-home care has worked well in the past and I would like to continue with that model.
The mutts are actually fairly easy to care for - but I need an experienced carer who understands their issues and who is willing to follow my regime. While the likelihood of an acute medical emergency arising during my absence is very small, I nonetheless have to have a carer who is prepared to handle emergencies if necessary. The job requires a bit more responsibility than many sitters are willing to take on.
The difficulty I have run across is that under the fairly new TschV dog care providers are required to have (or be in pursuit of) specific educational qualifications. Dog sitting or dog walking is not just a simple 'Nebenjob' anymore. Most of the sitters I've spoken to are unaware of the new law, or have no interest in complying with it - and I can not hire anyone who is working 'schwarz'. The few dog sitters I've found who hold the appropriate qualifications have not been willing to travel to my area.
(OT, but while I agree that people who take care of dogs should have training, I do think the TschV has set the bar unrealistically high - but that's a discussion for elsewhere.)
The insurance question is another compliance with the law issue - under dog control legislation in SZ (and most cantons) dog owners and 'those in charge of a dog' are required to hold liability insurance. In fact, the 27. Jan issue of K-Tipp has an interesting article on this - and I would urge every dog owner to read it. The article points out that if the person in charge of the dog does not meet the TschV educational requirements, one's insurer may decline coverage of any accidents that occur - leaving the owner in serious trouble should something go wrong.
I was surprised by another point in the article. Speaking to my insurance agent some months ago, I had been assured that a sitter working in my home would be covered under my Privathaftpflicht. But the K-Tipp article says this only applies if the carer works gratis; once payment is given the carer must have his/her own liability policy. I need to clarify that with my insurer - and if that really is the case, my idea of bringing the person from the UK here may not fly. I need to find out more about her insurance, and whether that would cover activities in CH, or if there is some kind of alternative policy available here.
My questions about permits are out of my ignorance. (I've lived here so long I guess I automatically assume that which is not verboten requires a permit.

)
Of course, I would very much prefer to have a local sitter - but so far my efforts to find the right person have been fruitless. Hence my attempt at looking for options outside Switzerland.
(Should anyone know of a person who is experienced, qualified, set up to work legally, and is willing to put up with looney toon mutts - please let me know. )
In the meantime, I'll keep exploring other possibilities.
Vielen Dank, mitenand.