Trumps voice and massive coast to coast influence in the White house would come with a fully operational and equipped television station. No? He would have ready access to nationally televised daily White House briefing that can be broadcasted nationally to the American people. No?
Now ad in Trump having total access to the entire file cabinet, the data, intel, backgrounds, financials, donors, investigations and entire history on the D.C. disciples of deceit.☺
Presidents who have dabbled in the amendatory process have done so behind the scenes. Lincoln actually engaged in bribery to procure the necessary congressional supermajorities to get the 13th Amendment past Congress and out to the states for ratification.
Trump could work behind the scenes if he so wished.
The president has a role in the legislative process because the Constitution grants him sign-or-veto power over legislation. It's customary for a president to get involved in the rough-and-tumble of the legislative process.
But Article V gives the president no role in the amendatory process. Within the federal government, that role belongs exclusively to Congress. Congress tends to get prickly about its prerogatives. A president who openly gets involved in the amendatory process via the bully pulpit would find himself dealing with an outraged Congress, and both parties would join in the outrage.
Fortunately it would never get that far. If President Trump suggested inserting himself into the amendatory process, the Speaker of the House and the Senate Majority Leader would sit down and counsel him thusly.
Mr. President, Congress won't brook your open interference in the amendatory process. There is a time to lead from in front and a time to lead from behind. You need to lead from behind, and let people on your side, both inside and outside Congress, lead from in front. If anyone asks your opinion on the proposed amendment, just say, "It's a congressional matter, and I'm staying out of it." That kind of diplomacy will get you the supermajorities you need."
If I read your second paragraph correctly, you are suggesting blackmail. That is also doable, but it also involves risks. That is something that has to be done out of sight and not by the president himself, lest it blow up in the president's face and lead to an impeachment situation. Generations later, some enterprising author would go to the Trump Presidential Library and come up with the secret history of how Trump's team got an amendment proposal got past a reluctant Congress.
I'd go with the bully pulpit used judiciously. Newt would be a great advisor on how to play that. The blackmail side of the equation is something I'd save for a last resort.