I think ping in ad-hoc mode may work, provided that you add a static arp entry for the address being pinged.
Another option - start hostapd if AP mode is available. This will broadcast beacon frames and will be usable with ping.
If monitor mode is available, you can also try some of those various tools which employ frame injection.
EDIT:
And of course I'm stupid, because the easiest way is to trigger scanning, which typically is active and sends a bunch of probe packets. I think requesting association with some unavailable SSID also sends a few probe packets for this SSID before giving up.