Split ticket connections are you friend here. Kacee had a good suggestion; I searched and found IAD-HAM for ca. $3,300 on your dates so you could fly e.g. IAD-MUC-HAM/HAM-ZRH-PRG and PRG-VIE-HAM/HAM-ZRH-IAD for USD 3,300 plus EUR 400 (overnights in HAM on the way back, which might be a deal-breaker but it seems like only the morning ZRH-IAH has P space).
PRG-HAM can be done non-stop if you don't mind skipping being on *A for that segment.