is your Tableau Server also using an AD account?
If yes, you should declare its account on your MS SQL Servers.
I Have MS SQL Servers and I have seen that the queries coming from Tableau Server are done using its service account (the one that is declared at server configuration).
Without allowing SQL Sever accounts, I do not see much more. My linked servers are done with SQL accounts, not AD.
I hope it will help, I will try to make tests on my configuration.
PS: In order to test faster, publish live connections
I'm not sure why it wouldn't work, but perhaps you could create a view on SQL server using your custom SQL and Tableau could connect to that?