I think, first of all, you should keep calm and not be too nervous. It's normal for you to have concerns, but I also hope you won't be too formal, because all you want to see is your brother's girlfriend, not your sister-in-law. It's your brother's girlfriend who should be nervous, not you.
When you meet your brother's girlfriend for the first time, you should first greet her politely and give her a sweet smile to show your friendliness and welcome. Everyone says that the first impression is very important. If you can do this from the beginning, you will definitely leave a very good impression on the other party.
Maybe when you meet this stranger for the first time, you will be a little curious to see what kind of person she is and why she can get your brother's favor, so you can't help secretly looking at this other person. However, I think you can be more natural, just like an old friend you haven't seen for years. You have to talk and laugh, so people won't feel uncomfortable and embarrassed.
You can sneak a look at each other's looks and clothes. But if you keep staring at others, they will feel embarrassed, which will be impolite.
When eating, there are actually many manners. For example, when dishes are served, don't move chopsticks in a hurry, which is very reserved. You can invite everyone to have dinner together, or you can wait for your brother to say hello before moving chopsticks.
At the dinner table, it is definitely not just for everyone to eat quietly together, there will definitely be chatting. Just when chatting, don't point at each other with chopsticks, or point at each other with your index finger with chopsticks. Because this is called "guiding the way by the gods", which is very impolite.
Also, when you are eating, don't make a bad chewing sound, because it is really impolite. If you don't pay attention at ordinary times, you should pay more attention when eating.
Imagine that everyone is happy to eat together, but you have been licking your mouth and making some indecent noises at the dinner table. How can others have a good and happy dinner and chat with you?