Cooper 3stage multistage compressor - integrally geared. There is an intercooler between stage 1-LP and 2-MP.
From what you describe, since there is no vibration on this shaft when compressor is offline, and that vibration occurs on startup only, I would guess intercooler tube leak. Water accumulates on propane side (shellside?) of this cooler when compressor is stopped, depressured and cooling water is still running through intercooler.
Does vibration on this shaft disappear after a few minutes of running ?