When I was a kid, my parents were pretty stingy - making my brother and I order burgers or pasta while they dined on steak and lobster. I hated it then and I've never done that with my kids. As the person eating the 'lesser than' meal, it makes you feel like crap.
However, now as an adult, my parents insist that on my birthday (or similarly, we are going out this week or next because I've also just completed my degree) that I order whatever it is I want. Drinks too. That's how it should be if you are the 'guest of honor'.
Your husband needs to understand that as much as it is polite that you don't take advantage of the person who is paying (and he's welcome to do that when someone invites HIM out), it's also in bad form and extremely rude to insist that your guest pay for their own drinks. Don't like it, don't invite them out. Period. If I was to invite someone out to celebrate and they ordered the most expensive thing on the menu plus drinks - oh well. I have to plan for that. If I can't afford it, then I don't take them to that restaurant. It's not rocket science. I don't think it's your daughter who is showing a lack of maturity - your husband needs to get some manners here. You might also point out to him that the damage he'll do to the relationship if he makes her pay is not worth $2. And he will do damage. Imagine someone telling your that you and your accomplishment isn't worth $2. I'd be pissed.