I'd go for enchanting the ammunition personally, thats the part that really matters. I'd build the PC as a "Rune-smith" or Alchemist using the Focussed Practitioner template (with a few extra 'gun-nut' related stunts) with Ritual: Crafting and a bunch load of Refinement stunts with the slots left open for extra enchanted items. The rules for potions can really easily be used for enchanted ammunition, whether its enchanted arrows, potion grenades or rune-laden bullets, it also quite nicely limits the number of special shots the player gets per scene, and can avoid complicating things by enchanting a relatively modern item.
As a plus it means that not only is the players ammunition supply limited, he has to invest crafting time to rebuild his stock, and if he continuously relys upon the "rolling Lore/Fate point to declare that he coincidentally has an appropriate bullet" shtick, well I'd compel the heck out of that, "Well, you've been sooo busy chasing these Black Court Vampires all over town you just haven't had time to get around to restocking your ammo...", on the other hand, if the player finds just having enchanted ammo a bit limiting, this allows him to build some enchanted items to supplement/replace some of his ammo (enchanted night vision goggles, enchanted spotter's scope, magical stealth suit...)
I kinda want to stat out this PC now...