You have a dual transformer system, adding a third transformer is not a good idea, running a C wire from your air handler is the proper way.
I have a same set up (boiler and a separate ac unit in the attic and just went threw all this getting nest a C wire). I’m
not a 100% sure, but I think adding another 24v to Rc or Rh will cause your HVAC trouble.
IF you had a typical furnace with air handler it would be one transformer connected to Rh… You could then easily connect another to Rc and C.
I’m in no way a HVAC professional, this is just something I learned trying to get a C wire to my Nest.