Password Protect Application Form

Hey there.
I’m building a site for a client and want to add in a password application form.
I can make the page password-protected, as is standard for wordpress, but on the “Enter Password” page, I’d like it to say something like:
“Don’t have a password? Please complete the following form and we will send you one ASAP”
…and then have a simple, three or four field form to be completed.
What’s the best way of achieving this? Because the page is password protected, it defaults to the php-governed password page, so simple creating a page with a form won’t work.