Rings don't work in wild shape? They used to in 3e and PF I believe. You really need to shore up that low AC in bear form.
They did not work in 3.5e (
alter self, "When the change occurs, your equipment, if any, either remains worn or held by the new form (if it is capable of wearing or holding the item), or melds into the new form and becomes nonfunctional." - Wild Shape refers to
polymorph which refers to this). They might have changed it in errata, I remember that they went a few rounds with
polymorph/wild shape errata.
They do work in Pathfinder (overview of the Transmutation (polymorph) subschool, "When you cast a polymorph spell that changes you into a creature of the animal, dragon, elemental, magical beast, plant, or vermin type, all of your gear melds into your body. Items that provide constant bonuses and do not need to be activated continue to function while melded in this way (with the exception of armor and shield bonuses, which cease to function).")
They do not work in 5e (
polymorph, "The target's gear melds into the new form. The creature can't activate, use, wield, or otherwise benefit from any of its equipment."). But on the other hand, 5e
polymorph (and thus wild shape) doesn't use your own hit points - you gain the target form's hp, and when those are gone you turn back into your own form with the same number of hp you had when you changed, minus any excess damage (if you had 40 hp when changing, and took enough damage to take you down to -5 hp in polymorphed form, you would change back with 35 hp).