The Russians are much more subtle than that. They will use it to apply gradual pressure beforehand, rather than pull it at the last minute before they strike. When it hits the fan, you want as few variables in the equation as possible.

The day they cross, guns blazing, some border we have committed to defend, it will probably be too late to use the blackmail. Either our response will be already programmed, like a tripwire deterrent, or our commander will be pressured to act in spite of his being compromised. His hand will be forced regardless of what he'd rather do, or wished he could do. And besides, if the President refuses to do the obvious simply because he is afraid of the consequences, it will only raise more questions down the line. They can always overplay their hand, forcing him to lock and load in the heat of the moment, even if it isn't the appropriate thing to do. After all, he does give the impression he is that kind of guy, impulsive and instinctive. In a crisis moment, you try to avoid introducing an element of uncertainty for yourself, but do your best to present one to your opponent.

The way they will use their shakedown is in a more gradual, or finessed way. They will use it to dissuade the CIC from making the appropriate preparations before the event, or to make him select a response option more to their liking, or even to induce a note of caution or hesitation or confusion. It will be designed and executed in a way that will make the response more difficult. For example,
rather than threatening to reveal their blackmail in order to stop a military response, they may use it if we are contemplating signing a mutual defense treaty with Lithuania or sending a brigade of infantry there, in a hope to dissuade us from doing so.

Russia knows that force is not the only, or even the best, way to get your way. Sometimes the threat of aggression works just as well in achieving long-term geopolitical goals. This is why they react so violently to moves on our part (like opening a base, or installing purely defensive systems) that present no direct threat to them. They understand that defensive moves strengthen offensive capability, something we somehow refuse to acknowledge.

They understand the use of bluff and bluster, swagger and tough talk. It gives them a way to get something for nothing, or at least with no risk. And if it doesn't work, they can easily back down and save face or military loss. In other words, they act just like Trump, but they are much better at it because they think rationally and strategically, not emotionally and tactically. And they also understand that in our irrational desire to make a profitable deal we will invariably give away the farm. We are greedy, they know that about us and they use it against us.

In my opinion (after all, I only have access to the same fake news and alternative facts you do), whatever the Russians have on Trump is really no big deal, it is the result of some perfectly legit and reasonable business or political deal that would have been quickly forgiven and forgotten if it had been made public--like Tillerson's past Arctic and Gulf of Mexico oil drilling negotiations, or Manafort's consulting for the pro-Russian former President of Ukraine, or Sessions' repeated meetings with Russian diplomats. But the Trump administration's effort to sweep these items under the rug and to distract Americans from their existence has signaled the Russians that this is something Trump wants to hide and they can use that against the Trump White House. The Trump gang is simply too dumb to realize it, and Putin is too smart to let it slip past him.