I am considering an outdoor wedding in March. My concern is the unpredictable weather at that time of year. I can make backup arrangements, but how do I handle this on the wedding invitations? Is it wrong to put a second location "in case of bad weather" on the wedding invites, so guests will know?
That's exactly what you should do -- plan on an alternate location in case of bad weather (or plan to have an extra tent on hand at the outdoor site), and then make sure your guests know the plan. It's completely appropriate to include this info on your invitations.
Wedding Planning Advice