I agree with Alcie and Spike. Four weeks is very early in your recovery, and any worrying symptoms you might be having are expected at this point.
I also could burp post surgery. I was told I wouldn't be able to by my surgeon, but could immediately in the hospital. I asked another surgeon in the practice who said that some people can indeed burp after surgery. He said that it's possible for the stomach to let out a little air without letting out any other stomach contents.
I guess I don't really burp--I can't do it on demand. It's more like my stomach burps itself every once in a while. Be happy you can burp a little. It keeps you much more comfortable with less bloating.
I never had a post-surgical barium swallow either. That practice varies from doctor to doctor. about a year later, when I was worried I had done something to injure my wrap, I had one, which showed no reflux and no more hernia.
Try not to worry...I was told here it takes 6 months for most of the healing to occur, and a full year for the rest.