In short, I can make the repeater work, so that's good.
But I'm programming the radios to transmit and break squelch on the same tones. The transmit-tone part works because the repeater activates, but the handhelds are picking up signal (and breaking squelch) from a tone-less transmission on the equivalent simplex channel.
I have the tonemode set to TSQL and the Tone Squelch at 136.5. This matches the CTCSS/DCS Dec and CTCSS/DCX Enc of 136.5 which I believe is the transmit and receive tones from the RT97.
My goal is to make it that on that programmed channel the handhelds are (a) able to activate the repeater and (b) only hear from the repeater.