I think you are suffering slightly from the popular compulsion to do everything in the regexp. Personally, I would have gone for the simplest thing that came to mind e.g.:

my $keep = substr( $_, 0, 2 );
$_ = substr( $_, 2, length($_) );
s/0+/0;
$_ = $keep . $_;
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and then also reassured myself that such a simple regexp should gain significantly in performance and should win the trade-off against having the extra substr operations which are cheap by comparison to m//.