I know you said you cant stain the door but i think this colour in a paint would look good too. A nice coppery brown I guess. As far as complimentary colours go I would say orange but I feel like one orange door would look weird unless there was some more orange elsewhere.

No school bus yellow but what about a soft yellow? Or maybe a lightened version of the siding color. If not, I think white would be fine. You can always add color with landscaping.

How do you feel about a contrast with aqua/turquoise? I think you could find the right color that would accent the blue nicely.