yes John's Relay diagram will do the job JUST if the relay really don't work with low voltage either... if does, will need a diode coming out from ballast or just right before the splice to feed the 86 prong to keep safe from the low voltage triggering the relay

Don't worry about ECU... is designed to work on the way it is and it has being in that way since ever

the death time between the switching is not critical, and is in fact unnapreciable. If was important, system wouldn't work