I dug around in the device console logs and found the following bluetoothd error messages that correspond with the times of the resets:
error 08:20:41.483385-0500 bluetoothd HCI expected event 14 with opcode OI_DHCI_VENDOR_OPCODE (0x0000FD4D) timedout, reason is STATUS 621
error 08:20:41.483670-0500 bluetoothd status 621, hlCb 0x00FE2B6C
The same code running on an iPhone X with 13.3 and an iPad with iOS 12 doesn't show the issue.
I'm wondering how broad this issue is likely to be as it is definitely hampering the performance of our app on this device. But I'm not sure if it's specific to my phone, all iPhone 6Ss on iOS 13, all older iPhones on iOS 13, etc.