Yup. This is a confusing situation to which neither manufacturer (MTH or Lionel) publishes a "solution."
You can choose to wait to see if someone else has a "better" solution but here's one that you can try that should only take a couple minutes.
1) Swap the red/black wires from your 3-wire Lionel crossing gates to the train transformer. So the RED wire from the crossing gate goes to the Black terminal of the train transformer. And the BLACK wire from the crossing gate goes to the Red terminal of the train transformer.
2) Connect the Yellow wire of the crossing gate to the NO (right-most) terminal of the MTH Realtrax ITAD. The MTH ITAD has a rocker/toggle switch which is set to the "track" position. This is not shown in above photo but presumably it is set this way already.
I believe this will solve your problem. If not, rest assured someone here on OGR will figure this out!
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Fine print: the "technical" reason why I suggest this is the MTH Realtrax ITAD when plugged into Realtrax switches the center-rail to the NO/NC relay contacts. But the Lionel 3-wire 6-14098 gates is looking for a traditional outer-rail trigger on its Yellow trigger wire. My idea is to swap the red/black power wire of the Lionel gate(s) so that the MTH ITAD's NO contact "looks like" an outer-rail trigger.