I'm not sure why they are writing letters for a number of reasons.
1st, isn't god supposed to harden the hearts of world leaders? If this is the case, wouldn't they be going against god's plan by trying to soften the hearts of one of the leaders?
2nd. They are supposed to turn on false religion first THEN the WT. They should have nothing to worry about. They haven't turned on false religion yet so this case will be thrown out without the need to write letters.
3rd. Isn't Jehovah in control? Aren't we supposed to rely on him and leave matters in his hands?
4th. Didn't Scientology do something similar? When the IRS wanted to remove their tax exempt they had each member file a lawsuit against the IRS. While they aren't filing lawsuits, they're asking each brother to write a letter.
5th. Trump likes Putin. Trumps son in law bought the WT properties. Jehovah allowed this?
If only they put this much effort into protecting children.