Good thinking, Brie.
Maybe he was hiding in Italy from fear of a master crook who he had previous links to? Did he have any previous links to Italy?
I've not seen much about his time in Italy, Sadie. If he hadn't had to seek official assistance for replacement papers we would probably never have known anything about him ending up there. At the time the Italian police didn't seem to have any previous on him.
The Shengen zone covers a huge area and Brueckner was able to cross borders at will. For the likes of him an absolutely ideal situation with his comings and goings being known to none but himself.
He claimed to have travelled by train to Italy via Switzerland. I find that absolutely bizarre given the number of vehicles to which he had access and/or his alleged skill set in lifting that which doesn't belong to him to be able to acquire one.
I don't think anyone will ever be able to find out for sure how Brueckner ended up as an apparent down and out in Italy, but it is a weird part of his story which kept him off the streets for some time.
"Timeline" is a word which features prominently in Madeleine's case. No-one will ever know what Brueckner's was.