Using a Small Signal Relay in Automotive Application
I was incorrect in one of of my assumptions in this question.

The input I am going into is an ignition input on an ECU. These are generally pulled down internally, which means you can use a MOSFET to switch the signal, since the drain will be at 12V and the source will be at ground.

I did end up using the small signal relay, and one of the surface-mount legs of the relay eventually broke due to vibration. My solution was to epoxy the body of the relay to the board, but in the future I would use a through hole relay.