tl:dr; write your gateway in Perl; use Net::IMAP::Server; override Net::IMAP::Server::Mailbox; and use one of the many Perl REST clients to talk to your server.
Your best bet for doing this quickly, while maintaining a reasonable amount of code security, is with Perl. You'll need two Perl modules. The first is Net::IMAP::Server, and here is the Github repository for that module. This is a standards-compliant RFC 3501 server that was purposely designed to have a configurable mail store. You will override the default Net::IMAP::Server::Mailbox implementation with your own code that talks to your custom email backend.
For your second module, choose your favorite Perl module(s) to use to speak to your REST server. Your choice depends on how much fine grained control you want to have over the construction and delivery of the REST messages.
Fortunately, here you have tons of choices. One possibility is Eixo::REST, which has a Github repository here. Eixo::REST seems to deal well with asynchronous vs. synchronous REST API calls, but it doesn't provide a lot of control over X509 key management. Depending on how googley your API is, there's also the REST::Google module. Interestingly, this family also has a REST::Google::Apps::EmailSettings module, specifically for setting Gmail-specific funkiness like labels and languages. Lastly, the REST::Consumer module seems to encapsulate a lot of https-specific things like timeout and authentication as parameters to Perl object instantiation.
If you use these existing frameworks, then about 90% of the necessary code should already be done for you.
Don't do this by hacking Dovecot or any other mail server written in C or C++. If you hack together a mail server quickly using a compiled language, your server will sooner or later experience all the joy of buffer overflows and stack smashing and everything else that the Internet does to fuck over mail servers. Get it working safely first, then optimize later.