But how far has this reference frame itself traveled during that one minute?'
Relative to what? Relative to itself, it hasn't traveled at all. And since we don't know the mechanism for time travel, there's no reason to use any other reference frame. Really, until we understand how they are supposed to travel through time we can't discuss the interactions of reference frames across time skips.
There's a reason we don't already use visible light signals to send wireless data (except if we're lost in the wilderness, I guess). It's VISIBLE. Can you image how annoying it would be to have light flickering around you all the time from your communicating devices? One of the primary advantages of the various bands we use (radio, infrared, etc.) is that they don't interfere with our normal operations: they're invisible.
We've got plenty of bandwidth that doesn't interact directly with the human body. Why don't we stick to that instead of trying to use something that does?
If the facts don't fit the theory, change the facts. -- Albert Einstein