I'm left handed and writing neatly is something I don't do as an adult. Actually, most of my peers have horrible handwriting, at least the men do.
I remember getting an assignment where points would be deducted if there were legibility issues. Of course, I received a few jabs over it. I turned in a flawless essay, panties dropped and everybody clapped. Okay, not really, but close.
So why does your boy have horrible writing? Ink. If he's using a pencil, he'll get a lead mark across his hand. Using a pen is worse. You get the mark on your hand, and you'll smudge all your letters as you write. If he can lift his hand, write in a way that makes sense, and make it look good, then lucky him. For me it was something I ignored growing up and it caused me a lot of discomfort, and luckily by the time it became a problem everything was going digital. Sure, he can slow down and write beautifully, but then it comes down to practicality.
Do you want to raise a neurotic perfectionist or somebody that manages their time effectively? I pray for the latter-I have a co-worker that confided in me that his father (a wealthy man in one of the worlds poorest countries) inflicted great emotional trauma on him growing up, with high expectations and coldness when he did not satisfy him. When he told me of his past and why he's a perfectionist even to a fault, he spoke both humbly and shaken. He's 32. Thirty-fucking-two and that trauma still rules his life.
Your son's writing shouldn't be a problem. He's left handed, just like you thought. Unless you have any other reason to doubt his motor skills you're seeing ghosts.
It really sounds like you're raising a young boy. Definitely listen to your doctors, but I need to represent the southpaws here.