Note that I have never used this API nor the web front end which you are currently using, so most of this is just general discussion. Using a web API instead of trying to interface with an HTML GUI front end would be better on many counts: supportability, efficiency, documentation (of the interface) and, so long as you adhere to the terms you shouldn't get arbitrarily blocked by the provider.

I was doubtful that we would qualify for access to the API based on how I read their requirements (it appeared to me that to qualify, a public project/app must be created).

That may well be the case, but why not make your code (at least some of it) public anyway?

The biggest reason I didn't think I could use the API was that it appeared they only had clients for Java, C#, and PHP.

The CPAN is chock-full of clients for APIs where the API provider hasn't seen fit to provide a Perl client. Don't let this put you off. And ask Google for a client anyway - if they realise there's a market they might be more inclined to supply it.

are you suggesting that it might be possible to interface with their API using a Perl REST module?

The document certainly implies that, yes. I tend to use REST::Client but there are other modules available.

If I were in your shoes I would definitely at least attempt to use the API before dismissing it out of hand. Good luck.