There are few perennial vegetables and beans are not among them except for a few types.

First, regarding surviving winter, up there in Michigan, most vegetables don't. About the only ones are parsnips, shallots, garlic, horseradish. Some herbs are perennial, like sage. Some self-sow (which makes them seem perennial!) like peppermint.

As for pole beans, putting them against 3.5' fence will cause them to spread out sideways once the vines reach the top of the fence. Most pole varieties can easily reach 10' and more (i.e., Kentucky Wonder, Blue Lake, Rattlesnake).

You can try Burpee's Golden Pole Bean, it's about the only variety I can think of which doesn't try to climb as high as most others. (Burpee themselves discontinued this variety a decade or two ago, but you can still find it from the occasional supplier of heirloom beans.)

You can also try a half-runner bean; they won't really climb, but if your dog is small enough they may form a sufficient barrier.

If you want to consider things in the flower department, there are all sorts of ways to cover a fence, ranging from roses to climbing nasturtiums.

How about Reines de Violettes, an old bourbon rose. The canes grow to at least seven feet and I stretch them across my corner fence in both directions. Rose colors are pink to purple depending on the cloud cover.