I do not use Timeshift, but I have a thought on your question.
Maintain two folders for Timeshift naming them “laptop” and “Geekom”. When you are using Timeshift on the laptop, rename the folder “laptop” to “timeshift”. When you are using Timeshift on the Geekom, rename the folder “Geekom” to “timeshift”.
You could probably also use partitions each with its own folder for “timeshift”, but this would probably be less flexible.