It's usual to redefine the constants using other constants. The most well known case is the fine structure constant, aka alpha, aka almost 1/137. http://en.wikipedia.org/wiki/Fine-structure_constant#Definit...

The idea is that in many particle physics calculations you don't use the charge of the electron alone. Every time it appears, it's multiplied by other constants like c or h, so you redefine it as a new constant that is the usual product you have to put in the calculations.

The problem with constants with units is that they mix real physics with the arbitrary choose of the measurement units, like the time the Earth do a complete spin divided by 24 by 60 and by 60 and other complete arbitrary chooses.

It's difficult to say if "e" the charge of the electron is big or small. But in many calculations you can use alpha that is clearly a small number (~=1/137) and try to use perturbation theory to get a good approximation of the actual result. (You can imagine this as a lot of Taylor approximations.) (There are a lot of technical details hidden in these calculations that can make a mathematician cry but a physicist happy.)