I'm pretty sure Picon Bière is in fact a distilled product, just one meant to be added to beer to indeed produce a shandy-like concoction. The label states "Pour mélanger à votre bière" and the ABV is 21%. That said, to order said concoction at a bar, you would also just ask for a Picon-Bière.
As far as replacements for Amer Picon, my favorite by far is Bigallet China-China, which is defined available in the States (even Virginia ABC carries it! )