If you give envsubst a list of variables, it only substitutes those variables, ignoring other substitutions. I’m not exactly sure how it works, but something like the following seems to do what you want:
$ export THIS=THAT FOO=BAR
$ echo 'dont substitute $THIS but do substitute $FOO' | envsubst '$FOO'
dont substitute $THIS but do substitute BAR
Note that $THIS is left alone, but $FOO is replaced by BAR.