I continue to poke at HLink billing. I have yet to have a rejected claim be successfully accepted. I sent (and have received a reply to) an email to my contact at HLink regarding when I need to change batch/transaction numbers and I “get it” now. I'm doing another test and if it passes (last test took three days for HLink servers to process; I'm not sure why it took so long… must've been down for scheduled maintainence) I think I have enough of the edge cases covered that we can start to turn our attention to the billing user interface.
I think I am simply misunderstanding the wording around when a new claim number must be used and when a new batch number must be used.