I have exactly the same random 4wd issue and unfortunately have for years. I'm not a great fan of the car when the 4wd light is on in the wet. Even in the dry it behaves a little differently.
Same thing. Pump runs not long after startup, interior lights dim and then toss a coin if the light comes on or not. At its worst about 90% of the time but lately about 50%.
I once had a theory that holding the brake on at startup might fix it. Maybe it delays it. (not so sure if it actually makes any difference but it makes me feel better).
Seems strange that it happens early after startup - if the 4wd light doesn't come on initially, it then won't come on at all.
Does anyone know if the 3 relays near the Attessa pump have anything to do with it? (I did fiddle with these once and just created a 4wd and Hicas issue!)
Offering a reward to the first person to solve this.