For future personal and Google reference...
Running tcpdump without any arguments results in this message on my machine:
tcpdump: packet printing is not supported for link type BLUETOOTH_HCI_H4_WITH_PHDR: use -w
Turns out it's just because, by default, tcpdump chooses the first interface it sees, which in my case is the Bluetooth one. Specifying another interface with -i is sufficient to avoid this message:
tcpdump -i eth0
(It turns out Google did not display an immediately useful first result, and Superuser has no entries concerning this message. It's maybe faster to use Google than to find the 17th man option, which is -i).
I should probably have added this as a self-answer in Superuser, but oh well...