I want to pass an entire s/foo/bar/i regexp substitution into a method as a (preferrable single) parameter.

Can that be done?

No, I don't think so. In your place, I'd pass two things: a qr// regexp, and a code ref for the substitution part.

sub foo {
     my($qr, $replace) = @_;
     s/$qr/$replace->()/ge;
}

$_ = "This is a new dawn.";
foo(qr/(\w+)/, sub { ucfirst $1 });
print;
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This makes use of the fact that $1 etc. are (localized) global variables.

Alternatively, pass just a code ref for the whole s/// statement, as a callback. This callback can do more than just a substitution, though.

sub bar {
   my $callback = shift;
   $callback->();
}
bar(sub { s/(\w+)/\u$1/g; });
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