I don't have direct experience with the Luna-Pro F but I expect there will be a resistor wired in series with the battery check button to limit the current through the meter when the battery check button is pushed. If the resistance has decreased (or worse gone short-circuit) with age, then this could cause the effect you describe. If the resistor is wired in series with the button as I suspect then you should be able to check the value with a multimeter with the resistor in circuit, while the battery check button is not pressed, since one side of the resistor will be disconnected by the open-circuit button. You could then compare the measured value to the nominal value determined from the resistor colour code, and replace if necessary.
Update: I found a circuit diagram for the Lunasix 3 at
https://farm9.staticflickr.com/8258/8686556384_f936f35ce3.jpg and it is as I thought - the 12k resitor in series with the battery check button limits current. Better still there's a battery test calibration variable resistor in series as well. If the Luna-Pro F follows the same general scheme, then you may just need to find and adjust the battery test cal resistor.
(Only saw Fred's reply after posting this. Adjustment may not be required after all. On my SBC it does not peg to the right and I assumed the F was the same)