I assembled another board and now the speaker works during voice calls! Previously it hadn't, and I couldn't figure out what it was for the life of me so I just made another one and voila, it works, so I'm assuming it was either the module or some random short underneath the module that I couldn't see.

Anyway, it's a fully-functional and voice-enabled LTE shield and I'm planning on launching a crowdfunding campaign to get this project off the ground!