My initial impression is that there is no way to salvage this design. In order to account for differences due to batch I would need to have included samples from conditions D/E in batch X and conditions A/B/C in batch Y - is that a reasonable conclusion?

Yes, having samples in both batches is the one and only way to correct for batch effect.

Additionally, I wondered whether if I just included samples from conditions A/B/C in batch Y whether I could batch-correct the data based on the subset of shared conditions? In this way, it seems the data is only partially confounded.

At a bare minimum, you need only one sample sequenced in both batches to correct for batch effect. It is best if you have more of course, but as we usually assume that a batch effect affects all sample similarly (no batch:sample interaction), it is statistically not required to have all samples in both batches.That being said, the more data you have, the best batch effect correction you can make.