How would you handle this scenario?
How would you do it with a web site? In HTML, you'd probably have a form, and using input controls you would collect the information that you need, and when the form was submitted the appropriate request would be created by the client and dispatched to the server.
From the sound of things, the semantics of this are unsafe, so you would normally use POST.
You could use multiple forms if you wanted to -- but multiple forms aren't better REST than using a single form.
Given that, how do you do it with a REST API? Same thing.
Is there a type of http request for this scenario that I'm not thinking of?
The way to figure out if there is a suitable HTTP method that has already been standarized is to look at the HTTP Method Registry. You can read through them all -- but more likely, look at the names, and follow up to read the reference to see if the semantics actually match what you are trying to do.
MOVE is defined by the WebDAV specification. So you could review that standard to see if the details of the semantics actually match your use case.
In practice - you probably aren't going to get a lot of advantage from the less common metrics -- they rather depend on having clients (and intermediate components) that were built with those semantics in mind.
POST has the advantage of being ubiquitous.