Maybe if we have faster than light travel we can assume that we'll have a faster way to transmit data. It's possible there will be a transitional period where we move at subluminal speeds and can transmit data at superluminal speeds, but I don't think being able to hit the speed limit ourselves will imply that we can break it with data.

Plus relativity will really fuck with things if we are able to reach the speed of light. Time dilation will reduce bandwidth from the perspective of the stationary observer because they need to sync up with a clock that will appear to get slower and slower. Maybe some kind of warp technology will avoid this, but if it doesn't, this will apply even if the signal can be transmitted instantly.

I've become resigned to the likelihood that if we ever do get out of this solar system, it will either be unmanned probes (like the ones we've already sent, which are still debatable as to whether we can consider them outside of it or not yet; they've passed the heliopause but haven't gotten to the Ort Cloud yet), or they will be effectively independent branches of humanity that will diverge and become their own thing over time. We'll be in contact (assuming both sides survive), but it will be more like pen pals where messages take years to arrive and meeting up in person is impossible. The ship will travel for generations before arriving at its destination with a good chance that they'll just die on the way or shortly after getting there unless they are prepared for further generations of terraforming moons and planets in the new system.

And with our current situations, even that seems unlikely. I'll be impressed if we see a self-sustainable moon colony or space station during our lifetimes or asteroid belt mining.