You need one answer from someone who does not report to your manager and does not owe you a slot. The draft that feels polite asks if they have thirty minutes this week. That draft competes with every other thirty minutes on their calendar, and it trains them to treat your name as a meeting.
The useful version is a note they can answer from a phone between two other things.
One decision, visible in the first sentence
"I need a yes or no on talking to Acme before we close the partner list, by Thursday 16:00 UTC. I recommend yes."
If you cannot close the list, you are not asking for a decision. You are asking for a think-aloud, and a skip-level is the wrong audience for that. Name the decision with the options included. Two options is plenty. Three is the ceiling.
A recommendation is not presumptuous here. They have less of the detail than you do. Withholding your read outsources homework they will not do, and the reply you get is a calendar hold.
Write as if they have none of the thread
They do not. They have a different set of threads, and yours is not among them.
Current state, the hard constraint, the stake if they say nothing. Cut the chronology of how you got worried. Cut the names of people they do not manage unless a name is the constraint. Respect their time by making a cold read sufficient.
Say what a sufficient reply looks like. "A one-word no is enough. A yes can be the word, and I will draft the outreach." People over-answer when you leave the shape blank, and a skip-level over-answer is a meeting you then have to attend.
A call is the fallback, not the product
Some things still want a live conversation: a real disagreement about the frame, a number that will move a lot of people, news that should not land as text. Those are rare. Most skip-level notes are a permission or a veto.
If you do need the call, send this brief first. The time is then spent on the part only they can do. Asking for the call in the first line, with the brief attached as homework, inverts that.
Say what happens if they are silent by the hour you named. "If I have not heard by Thursday 16:00 UTC I will not contact Acme and we will close the list as planned." A default they can live with is how you avoid chasing them, and how they avoid feeling chased.
How it actually goes out
You can write this yourself at your account after an emailed code, or have an assistant draft a briefing you then approve. Either way the words under your name are yours. If they have an account, you need an accepted contact for a new thread. If they do not, they get email and a page — nothing to install, and they can reply from the link.
Do not put a private nickname in the brief. They would read a word they have never been called. Use the name they use at work.
If the note is already sitting in a session, connect your assistant and cut it down to the sentence they can answer.