Why can't we tell, we are moving?

"To my knowledge, moving at relativistic speed means that time passes slower than for a non-moving observer. My question is, can't this be used to determine whether we are moving?"

Because there is no such thing as an absolutely non-moving observer. Everyone's moving in some reference frame.

If you see a someone moving at a relativistic speed relative to you, yes, you would observe their time to be moving slower. If you switch to their perspective of you, suddenly you are moving relative to them with the same speed and they're the one at rest. And the same rules of time dilation apply again (as all inertial frames are equivalent), meaning that they observe your clock to be moving slower now.

Since the same rules apply to both, so there's no preference here. If physical laws give preference to neither of you, then there's no experiment you can use as an evidence to say that there's something 'unique' happening in your frame that could qualify your movement to be absolute.

The above holds for inertial frames. People in accelerated frames can detect that they're accelerating because they experience pseudo forces.

In special relativity, there is a concept called relativity of simultaneity. It says that events that appear simultaneous in one reference frame might not appear simultaneous in another reference frame. In your situation, 1 and 3 will have synchronized clocks in the reference frame of 2. However, 1 and 3 will not have synchronized clocks in the world reference frame.

It is actually a postulate of special relativity that you can never tell whether you are moving at a constant velocity.

Your question implicitly assumes that there is a "absolute speed". There is no such thing. As far as 2 is concerned, they start at rest and then 1 and 3 fly off in opposite directions until again coming to rest.