The Bronco's switch is "supposed" to have power to the ignition during START, but if yours is acting up, that might be what's going on.
However, there should be a separate wire (at least there is in most aftermarket harnesses) for telling the computer that the starter is cranking. That's not necessarily the one that powers the ECU relay, but just tells the computer what status the engine is in.
If your switch definitely does not get power to the RUN circuits (Green w/red and Red w/green) in START, then a new switch might cure the issue.
If not, then there should be some way to pull power from the START wire (Red w/blue) to do the job. Maybe a jumper T'd into the main wire with a diode to keep the starter relay from energizing when you just turn on the key?
Probably not that simple, but at least it's a start for discussion purposes.
What chassis and engine harnesses are you using?
Paul