In general I would not expect an issue because if it is shorted, it will obviously be pulled low, an if it is not, there is likely enough capacitance to absorb the energy. But if the rail capacitance is very low or your device is extremely sensitive, then I can see your concern. It might be interesting to connect a FET probe across the power rail, and then switch on the resistance/continuity test.
I am not aware of any meter that has this configurable. Only dedicated low resistance meters like the Keithley 585 allow you to limit the maximum voltage a value like 100 mV (but not for damage, but to prevent breaking up any corrosion that may be on the contacts). Even in diode check the open circuit voltage will often be something like 6 V (just checked a Keithley 2000 and HP 34401A). Even if meter specify that the maximum voltage is 2 V, that is often only if the resistance is within range. The open circuit voltage will often be at least 5 V.
The only suggestion I have is to connect a resistor in parallel that is near the top of the range you are using. This should limit the voltage as long as that resistor does not get disconnected. That, or use a programmable current source / SMU that can limit the compliance voltage to a low value (or a programmable voltage source that can limit the current). That is how you measure resistance whenever the canned DMM resistance test are not sufficient for whatever reason (voltage coefficient, very low resistance, very high resistance, speed, etc).