Is there any logic applied between the transfer case switch and the actuator on the front axle?

Try unplugging the actuator at the front axle and see what happens. If you don't get warning lamps, that's promising. You can also try unplugging the switch at the transfer case and see what happens. I have no idea whether the system defaults to "axle disengaged" or "axle engaged", but obviously that's something for you to find out.

If the ECU is applying logic, it may have something to do with disallowing axle engagement if the road speed is higher than X, or some such thing, or possibly using a time-delay between transfer case selecting 4x4 and engaging the axle, to prevent (presumably) dog-clutch shift elements from being damaged by trying to engage at high speed while the other part of the shift element is stopped.

I'd be tempted to fake the system out on the transfer-case-switch side of it, because that way if there is logic involved, at least part of that logic would be preserved.

My 99 turbo diesel transfer case has a clutch similar to an AC clutch that gets the front driveshaft spinning before the thermal front axle engages. Thermal unit burned out long ago and is now a manual operation and being in four low without front axle engaged makes for better sharp turns backing into the driveway without the "humping up". Just don't engage the front unless stopped or already the transfer case is in 4wd mode. The thermal front axle unit failed at the most inopportune time possible, stuck in deep sand with license plate for a shovel :)