I fully understand the difference between a noun and a verb. Further, I fully understand that the word "sin" in Hebrews 4:15 is a noun and not a verb (at least, in English).
Here, we disagree. The use of "sin" as a noun does not (at all!) preclude the "action" interpretation. If I am playing outfield in baseball, and catch (verb) a pop fly, I have made a good catch (noun). The part of speech implies nothing else there, and it implies nothing else in Hebrews 4:15.
I don't interpret "could have sinned" from the noun "sin", nor would I if it were in the verb form. Rather, I take the whole phrase: "We do not have a high priest who cannot to sympathize with our weaknesses, but One who has been tempted in all things, as we are, yet without sin" (NASB). If the writer meant "yet without the sin nature", I'm reasonably certain that phrase would be in the text. It isn't, nor does the concept appear in the context.
You posit a high priest who was never actually tempted because He was unable to sin. God cannot be tempted with evil (James 1:13), but God the Son, while in the flesh, could be, and was, and the temptation was either real (meaning He could have sinned) or it was not real (meaning He could not have sinned). If the temptation were not real, then Jesus could have simply yawned at the devil in the desert. Rather, He was genuinely hungry, and He could indeed have told the stones to become bread... and eaten them.
I find the idea that Jesus "could not sin" both speculative and incompatible with plain Scripture.