So say you have all yellow at start of game granting emotional spectrum fear. And use ability 4 to swap from outside the game and bring in say a green. After assignment would you still have the trait?
Since I believe it says if ring battery and attached constructs are the same which also makes me wonder if you start with a green attached and assign it would that also reactivate the trait since now the ring battery and attached constructs are the same?

Honestly I have no clue but then you fall into the area of

1) Does the Power Battery only create same colored constructs but you can represent them by using other colors

2) Will you opponent/judge allow it since you don't have the proper colors

Unfortunately no. In both cases you would lose the Emtional Spectrum Ability. At all times all constructs associated with the Power Battery (not those brought in with the construct trait) must be the same color or the Emotional Spectrum ability turns off.

Check out this thread which fully breaks down the Power Batteries and how they work: http://www.hcrealms.com/forum/showthread.php?t=522466

1) You can bring in any colored construct with Power #4. Nothing prevents you from doing that.

2) This would have to be a house rule as per the rules if the colors do not match (doesn't matter if you own the corresponding construct of that color or not) then the ability does not activate.