Why do you need to use the push start? maybe something wrong with the ignition key system. Can't you buy just that unit? IMHO, oil pressure sensor pops up as your oil press is low. There could be some problem with the oil pump or maybe the oil pressure sensor as well..
The other day, my E32 cant be started but all other electronics are working but there is no engine light, brake light on the meter cluster during cranking. Checked the fuse and fuse 17 was burnt. A quick change and its back to normal. Fuse 17 controls the ECU if im not mistaken.