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Since the Canon 40D and Rebel XSi/450D onward, Canon has used an electronic first curtain when shooting with Live View activated. With LV Quick AF the mirror descends and the camera will phase AF as in normal operation prior to the mirror flipping up and the shutter opening. In LV Live AF or in LV manual focus the mirror stays up, the shutter stays open and the exposure is initiated electronically. Thus electronic first curtain has less vibration than normal operation with MLU, as in MLU the shutter has to open to initiate the shot, which causes some additional camera vibration at the beginning of the exposure.
In all cases the shutter closes to terminate the exposure. I know of no way around this. It sounds as if you would need to figure out how to use electronic second curtain if you have no shutter. None of the Canon dSLRs currently have that capability.
If you are shooting in LV the shutter automatically reopens again to get ready for the next shot.