You can't flip that seat over. One side is machined (or molded) to precisely fit the piston edge, the other side is not. Trying to pressurize the reg with the seat upside down wouldn't be much different than just connecting a 2nd stage directly to the tank....not so good.
I don't work on these too often, I don't own one, but the seat I'm looking at now has a small dimple on the side that fits in the seat retainer. But you shouldn't confuse this with any sort of recess to fit the piston; the two sides don't look at all similar, it't not like a flat puck seat that you can flip over to use the other side.
Rsingler was suggesting that rotating the seat, meaning turning it like a volume control knob while it's in the seat retainer, might subtly change the resonance between the piston and seat, quieting the whistle. Again, it's not really a 'whistle' its more like a harmonica. I don't know how much you understand about the causes of the harmonica sound (bet you're learning a lot!) but it's because air moves very fast over the piston edge when it opens up, and this creates vibrations. This is similar to how woodwind instruments like flutes work, and when conditions are right, there's enough vibration and resonance to cause the 'musical' sound. It's not a leak at all. If you have any sort of leak, your problem is different. A leak and the harmonica sound? That means two different (but maybe related) problems.
All the tricks like re-lubing the o-ring, moving the seat, bending the small spring (that's a good one, rsingler, BTW, and so is the check to make sure the main spring is not 'balanced') etc...are to disrupt the resonance that occurs with this quick air movement through the piston. It's not just a MK25 problem, all piston regs are susceptible to it. My MK2 sang like a nightingale after a rebuild once.
Anyhow, best of luck and hopefully the problem is solved.