Did you read the datasheet for this unit?
In my experience, battery voltage does drop while cranking
but it is usually in 9-10V range worst case (ok, that's on my car).
The power supply from the website you posted link has wide
input range 7-18V for 12V operation.
And yes, it does have low voltage detection and it is set at 11V but
this initiates shutdown if the low voltage condition lasted 10sec.
I don't know about your car, but my starts in 1-2 seconds.
I other words, this PSU should be able to bridge cranking periods
and if not, maybe you should have your car's electrical system
checked out.