I have a similar problem in a 91 e30. Car stars fine on the first try or after sitting a while. If its within 30 mins of going on any reasonable drive the starter will not start (or even try).
Which relay is the unloader relay? There is orange, blue, and white on the cowl and I forget which is which.
All electrical works in the car, when you go to "start" on the ignition switch the lights dim for a second but there is no attempt to start.
Battery voltage is at 13.7 while running which seems low but not enough to cause this. Before trying to start after sitting battery has 12.6V. Battery is only three months old or so. Using my battery tester it says I have good CCA available when I have the problem
Problem seems to be more common with a higher ambient temperature.
If I get a jump car starts fine like there was never an issue which makes me think its an amperage problem? I can also push start the car.
If I take a screwdriver and jump the big posts on the starter it turns over fine. Gets 12v to the starter post.
How do I jumper the ignition and bypass the switch? Which color to which?
Also on the solenoid posts, what should be the reading at those? I get 10v to the lower one and about .7v to the other one with the key in start position.
Thanks. Waiting downtown for my car to start

no room for push start.