As the title suggests, I am wanting to install a 15 amp double pole GFCI breaker, changing the current 15 amp double pole standard trip, on an existing 220v circuit ran with 10/2 Romex. Normally on a single pole 120v GFCI breaker, I would just put the hot/black to the brass screw, neutral/white to the silver screw, and then connect the pigtail on the breaker to the neutral bar on the main panel. However, with the 10/2 Romex only having 2 hots and a ground, how do I proceed, if I can? Is it safe/acceptable to connect the two hot/power conductors to the proper breaker screws, and then attach the pigtail to the neutral bar? DuckDuckGo'ing has left me with conflicting answers, and I figured somewhere here may be able to clarify. Am I just SOL without having 10/3 ran to the device? I appreciate any help!
You haven't said what your load is.
If your 240V load requires neutral due to including 120V loads, such as a dryer or oven, then you are out of luck. You will need to run 10/3 cable.
You must not misuse the 10/2 ground as neutral! It is not made for that, and it will mean your feeder is not grounded. A simple neutral wire break here will electrify all the grounds and kill you even if you have GFCI (RCD) - as discussed here. Note at 06:00 where the RCD (GFCI) just sits there laughing at you.
If your 240V load does not require neutral, such as an A/C unit, water heater, most shop tools, EV charger, etc., then you can use 10/2 cable. Since the white in 10/2 is neutral, you must wrap black or colored tape around the wire to re-identify it as a second hot wire. Black is fine; red is ideal.
When wiring a 2-pole GFCI without neutral, connect the GFCI pigtail to the neutral bar, but connect nothing to the GFCI neutral screw.
Yes, that is exactly what you do. Even though there is no neutral connected to the breaker, you still must connect the pigtail to the neutral bus.
Observe that if you connected this to a MWBC, with two hots and a neutral feeding two 120 volt loads and both loads where drawing exactly the same current, there would be no current returning on the neutral.