Not getting into the technical aspects of adding a third function to run your grapple. There are others here more capable to address those issues.
However, you mention possibly using the existing rear remotes to operate the grapple. My L has a dealer installed WR Long electric diverter to run the third function for the grapple. It’s operated by a rocker switch that’s under the operator’s thumb when palming the loader joystick. Being a diverter that’s on the curl circuit, I can manipulate the boom and curl OR boom and grapple open/close, but can’t operate curl and open close simultaneously. IME that setup works quite well for grapple use. Also have three OEM rear remotes with the valves in the OEM slots by the operator’s right hip.
Using a grapple on a CUT, generally the operator’s left hand is on the steering wheel while right hand is on the loader joystick and feet are controlling movement of the tractor. Even with being limited to two simultaneous functions as is the case with the diverter, having control of the grapple open/close under your thumb while retaining control of the joystick is a significant operational advantage to releasing the joystick to move your right hand to a lever elsewhere to open/close the grapple. If your tractor has a hand operated reversing lever under the right side of the steering wheel (instead of HST) your right hand has three places to go (joystick, rear remote for grapple, and reverser lever) to do any work with the grapple.
I’ve used plenty of equipment with more controls spread around than I have hands and feet to operate. It is far less than ideal, particularly if used many hours, IMO. Point of all that is you can save money using the rear remotes you already have, but you’d be sacrificing quite a lot of operational efficiency and ergonomics compared to a joystick mounted third function. As always, your machine, your $$, YMMV..,