With all due respect, the OP isn't asking what you like, but asking advice of how to use Tri-X while shooting at wide apertures in bright light. Incidentally, I agree with you that wide open isn't everything it is dreamed up to be, but that is way besides the point.
Sweet spot of the lens? I'm sure it's there, but depth of field is a tool that should be exploited to the same extent focus, brightness, composition, gesture, and treatment of light should, so to lock into a single aperture and calling it best is missing the point of having several apertures available to you. I mean, why do you think they are there? That's right, to give you a choice.
If you can shoot the same film in all lighting conditions, then everything becomes much easier at the printing stage, because you know what to expect, and in my experience that saves me a lot of wasted (expensive) paper. If you don't like Tri-X grain, that's too bad, but that is again not what the OP was asking about. It doesn't actually matter what film the OP is using, or what aperture they use, because it's a question that applies universally regardless of materials used.
I'm not trying to be a jerk here, but you are not focusing on answering the OPs question.
So, again, if somebody arbitrarily wants to shoot at f/2.8 because they like what their lens does at that aperture, using ISO 400 film, but their shutter can't open and close faster than 1/500th of a second. If you rate TX400 at 250, you could shoot at f/11 at 1/500, but if you really want to open to f/2.8 you're going to overexpose four stops, or similar to EI 16. Now you're starting to push what's reasonable for Tri-X to manage, so why not put a four stop ND filter in front of the lens?
It doesn't change how the film sees color, and it doesn't change contrast. It keeps all other things equal.
To me that sounds like the perfect tool for the OP's needs, and it does answer the question, while explaining why.