I think you should just raise your price and be done. Let people know that you were in an introductory phase and now have to bring your prices in line with reality. Let them know that you sincerely appreciate their business too. I don't know if I would go as high as the big competitor, though, as the price could be a point where you could gain market share. Instead, go lower and do better and promote that. By that fact alone, you can gain customers.

People know that if you don't charge enough, you won't stay in business. You have to get your prices up or you'll die a slow death. I say do it and get it over with.