Most people have missed the bit about time dilation messing up the clocks used in signalling, which I thought was interesting at first. However, surely the fix is just as simple as including a timing signal with the transmissions?
I think the problem is less timing the messages, and more that the messages from earth will just get redshifted more and more as the ship accelerates, which will require an ever larger antenna to pick up. This also has the affect of bandwidth tending towards 0.