It's not cheating. There many routes in the Sierra, and many destinations. There are only a few (Donahue Pass SOBO from Yosemite, Whitney zone) where a permit from one authority isn't good in another jurisdiction.

For example, I tried getting a permit a few weeks ago in the Sierra NF. I could get a permit easily, but explicitly NOT for any itinerary that included Yosemite. That is unusual, and the ranger apologized for it. You may find that Inyo gives you some specifics about Rae Lakes when you get your permit.

I often look at lesser known trailheads or routes to get where I want to go, and that includes permits for different entry points.

You can always call and ask. That's what those phone numbers are for.