The duration of a second is defined using Caesium-133 [1], not using anything else like daytime or rotational speed. It's independent of such factors and will always be the same in a given frame of reference. No need to update those values.

edit: I think it will lead to the same or more problems because everyone would have to change all time-related constants. Easier to add a second to the day constant (because the day constant is a subset of the seconds constant, but the seconds constant affects many more aspects than just the day constant).

[1]: https://en.wikipedia.org/wiki/Isotopes_of_caesium#Caesium-13...

But in that case, doesn't the "second" constant change (or the coefficient) and a metre remains the same length?

A meterstick would remain the same length, but the SI meter's length would change.