Thanks for all the tips and advice, it's very much appreciated. I think I will contact Song Sparrow and see what they have to say, Jasmin.
I guess I'll try this TP thing a couple more years and see how it goes. I'd hate to remove the plant but I think that the goal of anyone that plants peonies is to, at some point, see blooms. My co-worker says that his TP that dies back will bloom at random from year to year. It's my understanding that buds form on the woody growth from previous years, so I would wager that it flowers from buds that have formed near or under the ground. Thanks again, everyone.