Bash Parameter Expansion says that the variable (FILE in your example) must be a parameter name. So they don't nest. And the last part of ${param/pattern/replacement} must be a string. So back references aren't supported.
My only advice is to use
${EXT:+.$EXT}
to avoid adding a trailing dot if the file has no extension.
(Apparently UPDATE
Apparently back references may beare supported in ksh93).
So you could use something like
FILE="${FILE/@(.*)/_something\1}"