I don't think you did anything wrong but has your mode C encoder been recently checked? If it is valid, I think that you should have just generated a "proximate" traffic icon on matey boy's TCAS and then dropped down to "other" as he carried on with the departure. Additionally, below 1,000' agl most TCAS systems are TA only, so what's the panic? I also bet that an RA wasn't generated. And as his rate would have been well in excess of 1,000 fpm, I'd suggest that you should not have been considered to be "threat traffic". Unless his TCAS system is a bit suss. Suggestion for the future when down in the weeds, squawk (subject to approval, if this becomes a regular problem) only mode A. Then, with advice from ATC, you will show up but no RA advice would be isssed.
And the Dash driver sounds like he has bigger problems as well.