Even if you could only send only information across time you can cause all sorts of causality paradoxes.
Imagine travel back in time and telling yourself the winning lottery numbers.
Talking about Kerbal, how your system would prevent such exploit like the example I gave (sending a rescue mission before the kerbals got lost?)
Other example: you send a probing mission that goes many places and detect ore in one specific location. You come back in time and send a mining mission to explore exactly that location, with all the information about deltaV, orbital maneuver etc. You can arrive this second mission in the mining spot before the probing mission even detected the ore (supposing the probing mission travel to a lot of places). This is a huge exploit. How to prevent that?