Will lumbar epidural shots containing lidocaine mylocaine and prilocaine with a local anesthetic show a false positive on a urine drug test?

No. The drugs are not of the same chemical structure targeted by the test. But numerous claims of positive cocaine urine test after seeing a dentist or having epidural has been made in the past.
There are. accounts that it could show up as a false positive for cocaine on an immunoassay urine drug screen. However, you could request a GC/MS to be performed, which would definitively rule out the presence of cocaine.

The best answer to your question is its extremely unlikely and not supposed to, but in theory could happen.