It looks like CGI::Carp uses the package global $WRAP to decide whether to print errors to the browser. So the quick-n-dirty way to turn it off is:

$CGI::Carp::WRAP = 0
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To only hide warnings in your module, you could localize the effect, by putting at the beginning of each sub:

local $CGI::Carp::WRAP = 0;
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This will reset the flag to its previous value when that sub finishes.

Longer-term, you could write a patch to make this more formal, and submit it to CGI::Carp's author.