Also Regexp::Common would meet your needs.

use Regexp::Common;
my $parens = qr/$RE{balanced}{-parens=>'()'}/;

# ...

# my fix
$theParam =~ s/([A-Z]+)($parens)/&doFunc($1,$2)/geo;

__END__
Using your script, it will print 
Example 1: func A returns <(func B returns <(2)>)>
Example 2: func A returns <(func B returns <(2)>+func C returns <(func
+ D returns <()>)>)>
[download]