rf range at 433Mhz with prototype stripboard
I would have skipped the breadboard in the first place.  At these frequencies, it is hard to predict what differences there might be between such a breadboard and a real PCB with proper layout and routing.

That said, the main difference in range will be the antenna.  If the antenna is not self-contained, then the ground plane also matters a lot, since it's really a integral part of the antenna system.  For a self-contained antenna, like a center-fed dipole for example, there is no issue of ground plane and you want the lump of conductor that is the circuit to be as small as possible.

Your antenna looks like it probably requires a ground plane.

3 meters is quite short for a properly functioning transmitter and receiver at 434 MHz while sticking to the ISM band limitations.  Again, you can't judge RF issues like this except with the final board.  A breadboard is essentially useless.