"Executor" has a specific meaning. It's a legal term referring to the person who manages a deceased person's estate in accordance with that person's will. It's pronounced differently too, the emphasis is on the second syllable. Here is a definition with pronunciation.
"Executer" is pronounced with the emphasis on the first syllable. It means the one who executes. That may mean (arguably incorrectly) to execute a will as in the above. Among other meanings it may also mean the one who carries out the execution of a person, as in hangman. Many dictionaries don't even list it as a word (executioner is much preferred in the hang-man context), but here is one that pronounces it.
"Executor" is absolutely the preferred way to spell the legal term. Some (myself included) would say that the "er" would be incorrect in that context. If I were using it to mean hangman, I would use the "er" and pronounce it that way. But even in that context, some would argue that it's a made-up word.
For your purpose, I would endeavor to avoid it entirely by coming up with a different word.