Lancun, thank you for your well thought out post.
I believe that much of what you said is true in some situations. However, the opposite is sometimes true depending on what deck you are facing.
The part I disagree with entirely is that Rescue Energy can be worked around via Flash Bite or Toxic Fang. That implies that a previous attack left you with enough damage (~90-110) that it was possible to KO you by those means the following turn. But the fact that they need to waste a turn leaving you at high damage in order to play around it on the following turn plays right into your Seeker strategy. The fact that they are stuck between a rock and a hard place in the respect is a good thing for you. They either KO you and play into Rescue Energy, or they leave you at high damage and play into Seeker. Of course, a problem with this is that you can't keep it up multiple turns in a row. If you Warp out a damaged Gyarados and Seeker it, you can't get the Rescue Energy onto it before their next turn. But the fact that it worked once isn't too shabby.
Your next argument works if you're facing a deck like Machamp or Trainerlock.
Seeker will always be tremendous, whereas Rescue Energy is suboptimal, because not dying is better than dying, and Rescue Energy could be played around.
However, against Luxchomp or anything that OHKO's you, Rescue Energy is always tremendous, whereas Seeker is relatively suboptimal because you'll never get to use it on your Gyarados who will rarely be at half damage.
As far as handing them a prize, lets assume they're 2HKOing you because if they're OHKOing, Seeker doesn't help you anyway. As I see it, giving them one prize over two turns at the cost of Rescue Energy versus giving them zero prizes over two turns at the cost of two turns of Supporters used up is not clear cut as far as which one is better. If you need to develop your board, you'd probably rather get to use your supporters and keep healthy board/hand position at the expense of a prize, whereas, if it's the endgame when you need to win without giving up any prizes, Seeker will get the job done.
tl;dr - Rescue Energy for when they OHKO you. Seeker for when they don't. How many you run of either is up to you.