Be careful not to overthink this.
Traveling between cantons is not an issue. In the BSL cantons it is residency and visiting that matter - just passing through at 60km/hr is not visiting.
You just have to make sure your dog is allowed in the canton of your destination. And as above, many of the BSL cantons have an allowance for limited visits.
Emma presumably has a permit to keep her dog in AG - and so she and the dog may travel freely within the canton, subject to the restrictions given.
SG is not a BSL canton, so there are no cantonal barriers to a stay there.
Don't over think this! Really - life for Listenhund is already pretty difficult, let's not make it worse for owners by over-complicating things.
---
Emma, a suggestion to make your life easier - Argauer Tierschutzverein takes in holiday dogs.
I see that one of their 'Patentier' dogs is a pitty. That suggests to me that they are likely 'Listenhunde' tolerant. They, too, are an animal welfare organization as well as a Tierheim.
Have you contacted them?
https://www.tierschutz-aargau.ch/tierheim/ferientiere/
Again, hope a good solution is found.
We lived in Aargau for many years with dogs and I am familiar with many kennels in the area. I am surprised you are having trouble finding a solution. Could it also be a question of capacity? The kennels get very busy during the school holidays so one has to book well in advance. Which ones have you called?
Emma,
Since you mention that you have only recently moved to Aargau, have you signed up for your Listenhund course yet?
I bring this up because I have found the best place to start whenever I need help has been the Hundeschule. Whether it's discussion of laws, looking for information, getting recommendations for vets, specialists, gear, or holiday care, or simpy a springboard for ideas - the trainers and other class participants have always been a helpful resource.
A Hundeschule can be your entry into a community of like-minded folk - and with a listed breed, involvement in a community can make your life so much easier.
I would imagine the trainers who run the Listenhund courses will know which Tierheime will accept your dog, so that is also a good place to ask.