Mr. Bingley may be everything a young man should be—he's super nice, and he's super rich—but he is lacking in one thing that's pretty important in this novel: battle skills.
He wore a blue coat, rode a black horse, and carried a French carbine rifle upon his back—quite an exotic weapon for an Englishman. However, from his clumsy wielding of it, Elizabeth was quite certain that he had little training in musketry or any of the deadly arts. (3.4)
Elizabeth might consider that a deal breaker, but Jane doesn't seem to care: she can protect the both of them from oncoming zombie attacks. It's a good thing, too, because Mr. Bingley is a little squeamish. When Mr. Darcy kills the zombie-infected staff at Netherfield, for example, "Mr. Bingley politely vomited into his hands" (18.67). Yuck.
Like his Jane Austen inspiration, Mr. Bingley forgets about Jane a little too easily. This time, at least, the reason for it is that he thinks she might have been infected with the zombie plague, which is understandably a major turn-off. Plus, his friend and sisters have convinced him that Jane never liked him, after all.
All in all, Mr. Bingley doesn't have much of a spine—not when it comes to standing up to the undead or the living. Not a bad guy, just not a particularly interesting one, we guess. But don't tell Jane.