1.5 mm tape, like 406, won't kill your 388. It will technically wear all of the parts faster, but many people use 456 on the 388 (which is also a 1.5 mm tape) and it's not a big deal.
The 1 mm equivalent of 406 is 407. The one thing I would implore you to do first is check the sticky shed thread that is posted in this forum. Some Quantegy tape is no good and can cause problems for your deck, but there is a way to check against this by the date stamped on the tape's box.
This is the aforementioned thread:
https://homerecording.com/bbs/showthread.php?t=228067
-MD