We did something similar (embedding a model simulator in a tableau server dashboard).
Login was a bit tricky though.
In the end, we had two options:
1.) Make the Shiny app public, embed that and hope no one noticed or found it.
2.) Force users to use two accounts (once for Tableau, one for Shiny (connected to our company SSO for internal users))
It really wasn't a decision for us (had to go with #2).
If what was in the app itself was not sensitive in any way (ie, a interactive visual of public or user provided data), I could see going with option 1 though.