If he's already made the arrangements, I would just go along with his plan, even if it includes dinner. It sounds like your husband is trying to do something nice and loving.
You said he has been supportive along the way. Wouldn't it be fair if you compromised this once? I think it would cause resentment if you make it clear your weigh-in is more important than spending quality time with him. Also, I'm sure your nephew already made arrangements. If he's home from college, it is really nice that he agreed to sit for your child. Cancelling on him would be unfair as well.
This is a one-time event, not a regular occurrence. If the number is that important (and it really shouldn't be), just skip the weigh-in for a week. Who knows. You'll probably post an awesome number the next week.
Skipping a weigh-in one week won't mess with success. All you are doing is easing up on a Friday instead of a Saturday. Long term, being so rigid and not enjoying life will interfere with your success.
So enjoy the evening with your husband. I think you are so lucky that he went to the trouble to arrange a sitter and plan a nice night out. Thank your husband for how supportive he has been and how much it has meant to you. Then, a few days later, mention to him that if he arranges a sitter and date again, it would work better for you if he planned it for Saturday and not Friday.