Something that has been overlooked previously: your opponent chooses your maneuver for you if you reveal a red maneuver while stressed. Stay on target doesn't reveal a red maneuver, it changes your revealed maneuver to something else. Now, the rules do say that you can't perform red maneuvers while stressed, so we have two possibilities:
1) Assume that the "can't peform red maneuvers" statement is a consequence of your opponent always changing your dial to a non-red maneuver and making it impossible, not an absolute rule. In this case Tycho can keep using SOT every turn to change directions as well as PTL to take two actions. None of those stress tokens matter, so the combination of picking any direction and boost at PS 8 should let you outmaneuver pretty much anything at PS 7 or less.
or
2) Assume that "can't perform red maneuvers" is an absolute rule. In this case you use SOT to rotate your dial to a red maneuver, and then the game breaks. You can't perform the only maneuver you are allowed to perform, your opponent doesn't change it to a non-red maneuver, and you can't finish the rest of Tycho's activation. This is pretty useful because if you're ever in danger of losing you can just use SOT with a stress token and the game can no longer progress and you get an automatic draw once time runs out.
Since #2 is obviously absurd and would allow any ship with an EPT and the ability to generate a stress token to auto-draw a game I think we can safely conclude that #1 is the only possible answer. But is it worth it? Is the insane maneuverability enough to overcome the fact that you're paying a lot of points for a ship with only two attack dice?