How can I insert a completely custom registration form in place of the stock one

So I’m working on this site that in the past I had hacked into the core login page which is very bad. I’ve since created functions to do most of the customizations I had done. One thing remains and that is for the registration form.

On the form I’m also having to submit some data out to another entity and this is handled with custom JS so it can submit twice. Along with that, the form needs several other fields, some hidden, and other code to make it all work.

What I’ve done in the past is to go into the wp-login.php file and replace the registration from with something to pull in my custom form from an external file. So everything above and below the form in the code is there, then I inject the custom form code in between. Make sense?

What I’d like to do is to NOT mess with the core file any more and to just inject my form code in it’s place. I can’t find anything anywhere that tells me how to do this. All I’ve found is how to add to the current form, but that isn’t going to work with the added things I’ve mentioned and custom on_click code.

Anybody have any ideas on how I can do this from my functions.php file?

Thanks.