It turns out Intuit's documentation is incorrect (at least for its QBOv3 REST API).
At https://developer.intuit.com/docs/0025_quickbooksapi/0050_data_services/020_key_concepts/00700_batch_operation Intuit says the XML should be formatted in the way I showed in my first post, but it should actually be formatted like so:
<IntuitBatchRequest xmlns="http://schema.intuit.com/finance/v3">
<BatchItemRequest bId="1">
<Query>
Select * from Payment WHERE CustomerRef = '1933' ORDERBY TxnDate DESC MAXRESULTS 1
</Query>
</BatchItemRequest>
<BatchItemRequest bId="2">
<Query>
Select * from Payment WHERE CustomerRef = '290' ORDERBY TxnDate DESC MAXRESULTS 1
</Query>
</BatchItemRequest>
</IntuitBatchRequest>
I hope this helps others in the future! Thanks for all of the replies, they helped to narrow down the issue.