there you go throwing that "bobber" term around again.....
since traditional bobbers were flat track machines, flat track bars are usually the best starting point. I like superbike bars because they are not as high or akward looking as flat track while still maintaining that overall shape. The original bikes that bobbers were made from had stock bars that are what is commonly known as "beach bars". since they are wide and offer good leverage a lot of guys kept them for their race machines.
If you want to emulate that "bobber" look, use the bikes that would have been real bobbers (harley 45s, Indian scouts and cheifs, early preunit triumphs, etc) for inspiration.
apes are for choppers. since you are building an early chopper style bike anyway (despite you calling it a bobber) this might not be such a bad suggestion, but you will be trading that early vintage street/race bike vibe for that 50's early 60's "leader of the pack" vibe. Personally I think ape hangers are useless pieces of crap because if you ride any real miles the arm fatigue alone will make them uncomfortable, but hey the sacrifices you make for style.