Just spit-balling here, but it seems that T’ will have to be much smaller than T in order for the inner search to run each candidate program (multiple times for statistical guarantees) up to T’ to check whether it satisfies P. Because of this, you should be able to just run the inner search and see what it comes up with (which, if it’s yet another search, will have an even shorter run time) and pretty quickly (relative to T) find the actual M.
Just spit-balling here, but it seems that T’ will have to be much smaller than T in order for the inner search to run each candidate program (multiple times for statistical guarantees) up to T’ to check whether it satisfies P. Because of this, you should be able to just run the inner search and see what it comes up with (which, if it’s yet another search, will have an even shorter run time) and pretty quickly (relative to T) find the actual M.