Jag mår bra = I feel fine (or healthy). This is good and sounds normal.
Jag känner mig bra = Same meaning as above, but not as idiomatic.
Jag känns bra = My impact on someone else is making them feel good. It sounds strange and very self-centered.
Jag känner bra = I feel fine, as in I have a good sense of touch or something. It also sounds strange.
It's just not something you'd say in English in that context (or at least not something I've heard said). "I do fine" sounds like you're referring to something habitual, rather than how you are at that moment. "So do you usually make a lot of connections at these networking events?" "Oh, I do fine."