This will work. It used to be done all the time in the large RV's that were used as mobile showrooms for mechanic's tools (Snap-On, etc.).
Depending on the alarm control panel you are using, you may get various types of "trouble" signals if the panel doesn't see the AC signal. This shouldn't interfere with operation, but could be annoying.
With a single motion detector, current draw will probably be in the 20 to 30 milliampere range. Your car battery should probably be able to power the alarm for several days without a problem, but just to be on the safe side, you may want to put in a separate battery just for the alarm. This could be charged when the van was running.
Good luck with the project!