It strikes me (pun intended) as not very sound business on the part of Browning not having a webshop where parts can be ordered. Having to rely on distributors and dealers for parts is problematic in terms of the time it takes to fulfill (end-)customer needs and while it costs the customer more, it doesn't earn the manufacturer as much as direct sales would. Probably why the aftersales and stocking spares isn't high on anyone's itinerary at Browning. For that reason I'd be hesitant to buy one: Having a firing pin or a bead-sight go through a chain of three to five companies before its on my doormat isn't very appealing. It probably isn't profitable for the dealer either, unless there's a solid mark-up at your expense.
It may be that the dealer contracts prohibit Browning from direct sales to end-users, but if that is the case, why aren't the dealers able to ensure quick turn-around on standard parts? I order my parts straight from Italy and am much happier not having to wait for a dealer to follow up on < 30 quid transactions. And I'll make the joke before anyone else does: "Yeah, but that's because Italian guns need spare parts".