"But the Republicans are pushing to go much, much further, and actually choose sides. All this nonsense about giving "moral support" to the opposition."

We should choose sides. We should be on the side of any group of people who peacefully protest- meaning they have the right to do so.
We should against the side that uses "heavy security aimed at preventing new protests".
Is this actually a difficult point to understand?

"How in Allah's name are we at all qualified to pick which version of pure Islam should rule in Iran?!?"

Of course, this isn't our business.
Nor are we qualified/or somehow blessed, to pick who should rule in Canada, Mexico, Iraq, Japan, or any country, other then the country in which we are the citizens.

But we can expect and insist that country follow it's own stated laws- whatever they choose them to be.
And we can expect any country follow it's treaty obligations.

And what we are talking generally is the President of the US, and he is suppose represent the American people. He or she should articulate in the best way he/she can those values which American consider important. We didn't hire some damn lawyer, we elected a president, and he should be behave in a manner consistent with representing the American people.
Obviously some Americans will disagree or agree with whatever he/she chooses to say or not say- and they should be free to say so.

"It would be sheer folly to us to enter think about entering this fight."

We are in the fight, a fight that people anywhere should not be silenced about how these people wish to be governed by their govt.