Brueckner could well have been worried that his car could have been seen in Luz on the 3rd of May. So if The PJ checked the number plate it would not have come up in his name.
There was a black saloon car that was heading in a northerly direction up Rua Dr FGM at the time that Kate and Gerry would have been appearing and walking down the same road to the Tapas restaurant. A witness by the name of Joyce (surname, I think) with her husband saw it do a very hurried about turn and head off in the way that it had come. It was near Baptista supermarket. They thought that it was suspicious enough to report.
There is a PJ Report if anyone cares to look it up. I dont think that a number was recorded.
If you remember, I speculated on an earlier thread that this maybe was the abductor arriving with his getaway vehicle, when he suddenly saw Kate anf Gerry coming down the steps/walking the street en route to the resturant. He panicked and did a quick turn round in one of the little side streets?
If I am correct in my thoughts here, then he ceratinly would want to get rid of it pronto.