I reread this thread and one I started at ChemPros. I think that we will try 4-methylpiperidine in either DMF or acetonitrile and try to remove the adduct via silica gel chromatography in the presence of 1% TEA. If that gives problems, we will try DBU, as per Rolnor's suggestion. One person at ChemPros also suggested DBU, and he or she also said that he went ahead with his or her coupling without purifying. I see advantages and disadvantages to the no-purification approach. After our first try (which did not go so well), we have had some success purifying at the amine stage; therefore, I am reluctant to change at the moment. That having been said, I also contacted a group that made a similar vinyl sulfone dipeptide, and they did not purify after removing FMOC. There are many roads to the top of the mountain...
I tried cleaning some glassware with a white precipitate that might have been the polymer, and it was very difficult. The white mass wanted to adhere to the glass. Just an anecdote.