Mixing frame rates pretty much always leads to issues one way or another. It is something that I try to avoid at all costs.
While computers and files tolerate "odd" frame rates, television systems are simply not designed to deal with that.
If something is for TV then you really need to supply a tv spec file so I am guessing your PBS station will prefer 29.97?
I am in Pal land and we get the same thing happen here. People shoot 24 becuase they think it will be more "filmic" but then end up in a world of pain when everything has to become 25 for TV output.
You will see many posts here where people shoot with a 5D an AF100 and a GoPro all at different rates and codecs and then wonder why they have problems.
Cinema tools will do a good job of converting your footage but thats wont take care of the audio.
Nattress standards convert plug-in may also help you.