Space expands and contracts but time runs faster or slower. If we choose Einstein’s c as a constant then both space and time are necessarily variables. So, if space expands, then time must also "expand" or (I think you mean to say) accelerate. When time accelerates, our clocks run faster. I don’t mean clocks in the mechanical sense but in the broader sense that everything the natural world is a form of clock. When time accelerates, atoms move faster, wavelengths grow shorter, rocks fall faster, etc.

We can’t separate space from time and what to do with time as a dimension has always been a big problem. I prefer the view that we have four dimensions of space-time called height, width, depth, and duration. Time is a part of all four and not really a dimension by itself but there are many other valid points of view including using time as a fourth dimension.

Einstein’s c is the absolute of choice in GR and SR but we can also use time as an absolute while c and space are variables or we can use space as an absolute while time and c are variables but you are right that we can’t use both c and time as absolutes in the same model.