If the photon had a small rest mass, the SI definition of the metre would become meaningless because the speed of light would change as a function of its wavelength.
The SI Committee could not just define it to be constant; instead, they would have to fix the definition of the metre by stating which colour of light was being used.
Standard time is adjusted by adding or subtracting a leap second from time to time.
There is also an overall slowing down of Earth's rotation by about 1/100,000 of a second per year due to tidal forces between Earth, Sun, and Moon.
Obviously it would be more natural to attribute those changes to variations in the units of measurement than to changes in the speed of light itself, but by the same token it's nonsense to say that the speed of light is now constant just because the SI definitions of units define its numerical value to be constant.
But the SI definition highlights the point that we need first to be very clear about what we mean by constancy of the speed of light, before we answer our question.
Previously the metre and second have been defined in various different ways according to the measurement techniques of the time. If we look back to 1939, the second was defined as 1/86,400 of a mean solar day, and the metre as the distance between two scratches on a bar of platinum-iridium alloy held in France.
We now know that there are variations in the length of a mean solar day as measured by atomic clocks.
Unlike the previous definitions, these depend on absolute physical quantities which apply everywhere and at any time.
Can we tell if the speed of light is constant in those units?