At ~1 GHz it will not make a big difference.
A transmission line has a characteristic capacitance and inductance per unit length. The gap in the dielectric has the same inductance as the regular coax, but much less capacitance due to a dielectric constant of 1. Lets treat the capacitance as zero. This leaves you with an unmatched series inductor. The inductance is about 300 nanohenry / meter, or about 150 picohenry for your 1.5 mm gap. At 1 GHz, that 150 picohenry is about 1 ohm reactance.
There is a second effect as well. When fully mated, the center conductor ends up with a uniform thickness, as the back of the pin is the same diameter as the socket. If you don't mate the connector completely, there will be a short area where the pin narrows before reaching the socket. This increases the inductance slightly. The inducance is logarithmic in the ratio of the conductor diameters, so this is a relatively small effect.
As you can see, that gap really matters if you are running at 10 or 20 ghz, where the mismatch caused would be many ohms. But at 1 GHz, it isn't likely to be a big deal unless you are trying to do precisely calibrated measurements.
I wouldn't worry about customers over-tightening an SMA connector, at least if they are doing it by hand. If you have ever used an SMA torque wrench, you will see that the design torque is higher than you can apply by hand. If they are using a wrench that is not a torque wrench, you should stop them. You should only tighten an SMA connector by hand or with a properly set torque wrench. Also, despite some persistent myths, tightening to torque is not required to meet signal integrity performance. Hand tightening is sufficient to reach the rated performance. Tightening to torque is only helpful for vibration resistance, preventing the connection from loosening over time.