How's the graft doing now? I attempted a similar graft of peyote onto bridgesii but it failed to take. Ime, PC pachanoi ('pachanot') makes a far better grafting stock, although what you have there as a "bridgesii" looks nothing like mine (too many spines per areole, and a lack of longer spines). I'd be inclined to say it might be more in the direction of a macrogonus/peruvianus although I won't lend that as much confidence as the previous statement. Was your specimen - perchance - mislabelled? This may work out in your favour with the graft.
The button is still soft, actually softer than before and very wrinkled. It is firmly in place, so i can't pull it off without force, but i can see slightly under one of the sides and it looks like a root is starting to form on the bottom of the button.
It certainly doesn't seem to be taking in any water so in the long run i might end up pulling it off and rooting it in soil. If the root starts to grow out the side then i will take that as a sign. The fact that it is firmly in place makes me a bit nervous to make any decisions for now.
As for the rootstock ID, it was not mis-labelled, but turns out i am an idiot

I just checked the pot and it is clearly labelled as "SP" (meaning san pedro, which is how i labelled my Pachanoi), so you are correct, it is not a Bridgesii! It should be a Pachanoi, but it is possible it could be a Peruvianus as i planted Pachanoi and Peruvianus at the same time. They were in different trays, but as we have established, i can be an idiot sometimes. I have no idea why i thought it was a Bridgesii, clearly i was rushing around and not paying attention! The good thing that came from this is you have taught me how to tell the difference, so thank you!
What is the result of yours? does the button still stick to the stock? My gut tells me that means it has succeeded, but this is my first ever graft so i have no idea.
I will try to get some pictures up soon.