Part of me half-wonders how things at the hospital would go if the radscorpion injured Lucy instead of Cooper. I have to imagine that Cooper obviously would use the last stimpak on Lucy, but then he'd have the matter of having to cut the stinger out of her leg. Then he'd likely have to hightail her to Camp Golf in hopes of finding her antivenom while trying to avoid the Legion. Beyond that, I haven't an idea.
This is gonna be kind of a hot take that sounds worse than it is, but I think that entire scene wasn't actually about the Radscorpion. I feel like that scene in particular came about as a way to insert Cooper's monologue about Theseus' ship into the episode in a way that still felt organic. And it also gives Lucy a reason to get caught by the Legion and further that plot.
And I say that fully loving that scene.
That analogy has become more popular in media over the past decade. Vision had a similar argument with 'White Vision' in I want to say it was Wandavision most recently to my memory. (Great series, btw). But Cooper really put a great spin on it in a way that made it relatable and understandable to people who aren't exactly philosophical. And being a writer myself, I can totally see where a writer might, in a fit of inspiration, wax rustically poetic about the ship of Theseus and then have to fit the scene to the material because it was just too good to scrap, but not exactly the type of thing you'd easily imagine a 250 year old irradiated cowboy giving a shit about out here. But then we remember-
He's an actor playing a bit after the end of the world in a reckless bid for identity after dissolution.
And that seems exactly the type of thing an actor would do- monologue to the audience while cutting poison out of himself. Dogmeat, obviously, is the stand in for us as the viewer- an apt device as she can neither talk nor influence the characters in any way- just like us.
It reminds me of Hamlet almost, how this small betrayal preceded and rather foreshadowed the later one. We have an actor who was once a king in his world and is now obsolete in this new one. Betrayed by the people closest to him. Haunted constantly by the ghosts of his past. Always searching to add meaning to his suffering. As Hamlet sought meaning from Yorick. Both touched by death. But I fear I've gone off course 😅
So to answer in a roundabout way, I think he would have eventually given her the stimpak, yes. In fact, I could see Cooper delivering that same monologue standing over her while she writhed in pain on the ground, then giving it to her only after he'd made his point, but would it have changed the tone of the scene? If he had to cut the stinger from her, would it have lessened the impact of taking her to the NCR outpost later when we find out he actually stayed? When he could have so easily left her there. Especially after they seemed still at odds after she failed to shoot the rope at the signal.
I think he would have given it to her, but he likely would have left her there and that would have messed up all of their scenes after because the journey to NV is where he started developing real feelings for her. Without that, Lucy might not even have become a prisoner of the Legion and that was crucial for her to realize, "Okay, maybe I am safer with him than on my own." And all those other delicious little choices that set up the betrayal in such a devastating way.