Yes, the character may do four attacks:
With Haste the character has 200% activity. The character may now do e.g. a Full Attack with 100% activity in the Normal Action Phase, attacking with both his TWC weapons, and then another Full Attack with 100% activity in the Deliberate Action Phase, again attacking with both his TWC weapons, thus having four attacks in total.
Since a Full Attack requires only at least 60% activity in a round, the character might also spend his 200% activity doing a Full Attack with 80% activity in the Snap Action phase, attacking with both his TWC weapons, then do another Full Attack with 60% activity in the Normal Action Phase, again attacking with both his TWC weapons, and then a third Full Attack with 60% activity in the Deliberate Action Phase, again attacking with both his TWC weapons, for a total of six attacks! Of course these attacks will all have a severe penalty since not the full 100% were used for the attacks, so that this option is only feasible if the opponents have very low DB, but theoretically it would be possible.