Update!
I bounced this question off of our Meraki rep, and Support as well. Support also said there's no way to get this info; However, our Meraki rep clued us in on what is apparently a well hidden feature. There's an ability for Support (whether they know it or not) to enable configuration of SNMP traps from the dashboard for login attempts.
They can toggle that on and expose some configuration options (which you control) that will allow the Dashboard to send SNMP traps to your collector's public IP.
I'll follow-up once we've gotten everything configured and seen what it can do.
Login Attempts page with new option after support enabled it:
Trap configuration options (v1/v2c also available):