usage - Verb have in experience

The example makes sense, but it is rather odd sounding. It means that the lady's being disrespected affected you and was a bad experience for you. Other such "have" sentences are completely normal English. For instance,

"As I was waiting in line, I had a piece of the ceiling fall on my head."

It is certainly not a causative "have" construction. It's been called "adversative" or "experiential" have. Other examples are "I have a hole in my pocket", "He had a truck put a dent in his right front fender" (this is also interpretable as a causative).

The two oddities in your example that keep it from being fully grammatical standard English are (1) the "have" is in the progressive, (2) the subject of "have", "I", is referred to in the remainder of the construction after "have" only by "next to me", which doesn't make much of a connection between you and the lady's sad experience.

For (2), looking back at the previous example I gave, "I have a hole in my pocket", note the "my", referring back to me. The "my" could be suppressed, but it would still be understood. "I have a hole in Louise's pocket" is an entirely different construction.