No-one's actually told you what the problem is. In redefine.pm, you redefine module::function. But, when you call function(), you're calling the exported version. Exporter does a glob assignment which is different from a reference. When you assign to the RHS of a glob assignment, the connection is broken.

The proper solution is for redefine to be a facade over module. Bring all of module into redefine, then redefine what you want to change, then everywhere you had "use module", change that to "use redefine".