It should be obvious that the "c" here was used as a magnitude not with units. Saying that's "failing dimensional analysis" is in itself grossly sloppy - since it totally misses the context.
But dimensioned quantities are meaningless without units. are you saying c has a value of 1.803e+12? Or that it has a value of 1? There’s a big difference between using furlongs per fortnight and speed of light as your units for speed. By saying “1/c s” rather than “1 m/c” he was just plain wrong and certainly in no position to be smug about it.
That’s like being a grammar Nazi even when someone writes something otherwise totally clear.
And it’s obvious in this case that the magnitude of c was in metric units.
And how do you know if he was even referring to the speed of light by “c”? I could call that a dumb assumption. Because it’s obvious he was using it in a non standard way “magnitude of the speed of light in m/s”.