"being honest" sounds wrong to me in your example because it suggests that the speaker is about to tell something that they were being honest about at the time of the events in the story.
See this example of expected usage in a past-tense story:
I struck up a conversation with a man on the street and being honest I told him his haircut didn't suit him
Perhaps the past conversation may have been "to be honest, your haircut doesn't suit you..."
And now, "being honest" is used during retelling because you were "being honest" at that time.
In your example OP:
We struck up a conversation and being honest, I was not sure he was homeless.
I absolutely expected the bolded part to be something he said to the homeless person at the time.
After reading the bolded part and realising that that didn't make sense, I had to reset my understanding of what "being honest" was used for.
It's unidiomatic (at least ambiguous) to use "being honest" in OP's example: even though "being honest" may be short for "if I'm being honest" (a shortening I don't see very much in the first place), its placement within this recitation makes it more suitable, and more likely, to be interpreted as a past gerund before anything else.
Because "being honest" is used where a past gerund would idiomatically appear, and conforms perfectly (e.g. the first example in this answer), it suggests the directly succeeding part of the story will pertain to the past.
Therefore its usage in your example is subtly jarring, and "to be honest" or "if I'm being honest" would have been better.