I would not equate the notion of a change of c with time with that of a
change of the conversion factor between calories and joules. One can define
two independent units for length, for instance, inch and centimeter. It then
becomes an experimental problem to find out how many centimeters fit into
what has been taken to be an inch. This is a laborious experiment but it add
nothing to our understanding of nature. Therefore, people decided to have a
single unit of length and define all other lengths in terms of it. For
instance, one inch = 2.5400000000000000000000000000....cm, exactly. Before
we knew that heat was another form of energy, we had separate units. Now we
know that calories and joules denote units of energy and so the calorie has
been defined in terms of the joule, one calorie = 4.184 joules, exactly.
However, the constancy of the speed of light (independent of the speeds of
the observers) is a true indication of what goes on in nature. The factor
2.45, 4.184 can never depend on time. But c can, and whether it does or not
becomes a true experimental question. Of course, at a particular
cosmological time, ct does have the units of length and is combined with the
space variables in the spacetime of Minkowski.
Moorad