He only brought it up during an argument? So he was trying to be hurtful. Now it has left you with even more anxiety. What age are you both? Three months isn’t a long time, unless you hang out together every given minute of the day. There’s still a lot of learning about each other to do.
I think it’s unfair he would say that to you in a fight and still choose to hang out with them over you after. If the relationship was longer it would be easier to say what’s right and wrong here. Maybe a talk with him to discuss each other’s feelings would help to see where to go from here

I say...Less unwritten/inherent expectations, more written/verbalised expectations.
It seems to me like this has little to do with pride or embarrassment, and more to do with communication.
If you feel neglected, tell him where/what/why.
If you feel you want to be invited to go out with his friends to the arcade, ask him.
Just communicate more, don't expect him to know or follow-through on what you have in mind, express your thoughts/feelings more openly and transparently.
Similarly, if he is embarrassed about you in any way, he should express those sentiments with you.

I agree with @Miram. Host them over for dinner and take it from there.

I'd ask them to come over for dinner.

No real boyfriend goes out without bringing their gf with unless you chose not to. If that's not the case. It's been only 3 months. Better drop the man now then put up with the same bullshit. How could he not want to take you.

Have a bar b que and invite them