Dear madam or sir,
This bid is just per se because I don't know what stack of technologies You use for website.
But, Your minimal budget is more than enough so I would like to apply for this job.
Work that needs to be done isn't that hard. I estimated 4 days just to be safe, but I could design and developed form for a one day and next two days work on integration with rest of the site (cases You mentioned) and testing. It really depends on what You use and in what state is your web page (I need to see source code), but I could work for a smaller budget (cca 100-150$) because my working hour isn't that high. Also, I've ability to engage other developers and designers if needed, no extra cost of course.
I hope You will find my proposal suitable and if You have any kind of question, feel free to send me a message.
Kind regards,
Matej Basic