With the new but not tremendously sturdy connector removed from the RT-351, and my tea still hot enough to smelt iron, I moved around to the passenger side of the car and removed the connector from the SURF, which involved also removing the attached 1.2m whip antenna.
With the connector removed, I refitted the connector from the RT-351 onto the SURF, and then moved back to the drivers seat (tea still too hot to drink without major oral damage) and installed the strong and rigid connector from the SURF onto the RT-351, using the nice hex bolts and crinkle washers from the SURF unit.
And finally, I refitted the 1.2m whip, onto the RT-351, and checked the operation of the connector. It now requires more force than needed to bend the antenna at its sprung ball joint to turn the connector. Hopefully now, when in use, the antenna will stay upright! And all completed during my tea break!