This is for anyone who may run into this problem in the future:
After speaking with a Google representative, the short story is that you just have to wait.
Initially, the same representative informed me that, even with test/beta orders, the transactions would process (just not charged). Because of this, I was under the impression that my code was buggy.
Turns out, beta orders are never processed and remain as "pending" for their entire lifetime (which is a couple of weeks). Real orders, on the other hand, could take 6+ hours to process.
In regards to how the product is delivered, that did not play into the processing equation at all.