Once you have sorted out your permission to reside in Switzerland*, do make sure you understand Swiss law on dog registration.
You will need to import your dogs officially. I have never encountered a tourist who just 'stayed' with dogs, so if you do not leave when your tourist permission expires you would in the first instance need to contact Swiss customs and the cantonal Veterinäramt to find out how to regularize your dog's official entry into Switzerland.
Do not underestimate how important this is! If a dog is not imported correctly, even if all other vax and paperwork is in order, the dog risks seizure and quarantine, worst case euthanasia. Switzerland takes a very hard line on import violations, as we have a huge problem with dog smuggling, and we are trying to keep disease at bay.
Obvously the easiest thing to so would be to leave when your tourist stay is up, then come back officially. under whatever permit you qualify for, and officially import your dogs at that time.
Once you are officially resident, you will need to see a vet within 10 days to get your dogs registered in AMICUS, the federal database of dogs (and dog owners). In Switzerland every dog must be traceable from birth/import to death/export. Another reason for making double sure you correctly import your dogs is that the veterinarian who does the AMICUS registration is a mandatory reporter; if he or she finds an irregularity the vet must report it to the Veterinäramt, with consequences to you and your dogs.
You must also register your dog with your local commune, and pay the annual dog tax. If you have not yet found a permanent place to stay, you register at the commune you are currently in, then de/re-register when you move again. Dog tax varies by canton and commune, expect something in the order of 100-200 per dog.
Be aware that most cantons require a dog owner to carry liability insurance. The policy amount will vary by canton. For instance, in TI it is 3 million. This insurance is often rolled into your Privathaftpflicht, but you must make sure it is there. If not, you can take out a separate policy, it is inexpensive, something under CHF 100 per year.
Coming from the US, i assume you do not have a pet passport? Once you are officially resident you will need to get this, your vet can organize it for you. A pet passport is needed to move more easily around EU countries, should you plan on traveling with your dogs. This document is also where all rabies and other vaccines are recorded and stamped.
If you need help with any steps in the process, please ask. There are many threads going into detail, so I won't repeat those here. You can read about AMICUS here:
https://www.amicus.ch/Account/Login
And more about official importation for a resident dog:
https://www.blv.admin.ch/blv/de/home...frettchen.html
And finally, be aware that BSL is a cantonal matter, about half the cantons have some form of breed specific legislation on their books.. You mentioned Tessin - is that where you currently live? If so, be aware that a German Shephard is on the restricted list in Tessin, and requires a permit to live in the canton.
You can read more about the various cantonal dog laws in this summary from Tier Im Recht foundation:
Tessin:
https://www.tierimrecht.org/de/recht/hunderecht/tessin/
Overview of all cantons:
https://www.tierimrecht.org/en/legal...onal-Dog-Laws/
ETA:
In Tessin a restricted dog may stay for 30 days if visitng as a tourist. How long has your GSD been here? If longer than 30 days, you need to understand what steps you have to take to get a permit for your GSD.
You would need to contact the TI Veterinäramt for info on how to apply for permission to keep a GSD in the canton.. The relevant legislation is Art. 13 Regolamento/TI). I don't read Italian, so can't help further than that. But DeepL wll give you the gist.
FYI, This also applies to folks who own holida homes and long stay tourists.
If you are not in Tessin, please let us know which canton you are staying in now and we can help with other cantonal laws. (GSDs are not on the restricted lists in any other BSL cantons, btw. TI's list is more extensive than most cantons.)
But do follow up on this, for your dogs' sake.
---
If I undertstand your story/timeline - honestly, it kind of sounds like you put the cart before the horse in making your move here. A non-EU person is expected to apply for, and be granted, a residence permit before moving here with their pets. The process, while not onerous, must be done to the letter of the law. For your dogs' sake, don't get this wrong!
If you need help with the paperwork, there are companies who do that. One with a good reputation is Moving Animals, based in Zürich.
---
*I'll repeat suggestions others have made - if you are applying as UHNW individual who will not work in Switzerland, this really is a 'have your people call their people' situation. Such permits are granted on an individual case basis, and I have heard anecdotes where interested parties with similar profiles were given wildly different responses from Officialdom. Suffice it to say, this permit is indeed for the UHNW. Mere HNW often does not cut it. But as always in Switzerland, YMMV. As Mrs D suggested, contact one of the companies catering to the ultra wealthy for assistance in connecting you with the right people to facilitate this.
---
Lastly, here is a document every dog owner in Switzerland must read, the Codex for Dog Owners. This is put out by canton ZH, but the etiquette and expectations described are applicable everywhere. I like this document because they lay it all out concisely, clearly - and in English.
https://www.zh.ch/content/dam/zhweb/...12englisch.pdf
Again, wishing you and your dogs all the best.