Copying/imitating wireless garage opener radio signal
The problem you're likely to have is that just about all modern RF remote controls will use a rolling code, so if you record and playback the same code it won't work because it will expect the next sequence number. It's meant to avoid a third-party being able to easily record and play back the code.

Your only practical option might be to get another remote control from the original manufacturer (or maybe you already have a spare?) and control that using the Raspberry Pi. A simple way might be to connect mechanical relay contacts across the button contacts to simulate it being pressed.