Your letter sounds like you've almost talked yourself into believing you're ashamed of what you want to write, just because you've had the idea of using a pen name for it.
#95: The False Competition Between Fanfic and Original Fiction
What's depleting you isn't the act of writing; it's the shame you feel about how and what you're writing.
#69: Getting Unstuck from “Should”
Every single approach to writing is perfectly okay, as long as it's in line with the kind of writer you want to be.