Hey there! :)
FullCalendar's documentation indicates that they already have some support for timezones under the hood; I'd imagine that implementing a few dropdown boxes to control that should be fairly straightforward. I'm curious though: is the site that we'd be working on publicly available that I might be able to take a look myself? Either way, I'd be happy to help.
As for the standard "about me" formalities, I've been using JavaScript both professionally and as a hobby most of my life, and I've been freelancing as an expert in JavaScript for quite some time now. By all means, feel free to check my profile for a few testimonials from past clients, as well as to get to know a bit more about me.
With that out of the way, I'd love to hear from you as soon as possible. Toss me a reply so we can get started right away! I look forward to working with you. :)