The whole point of dimensional analysis as a method of error checking is that fudging the units doesn't work. If you have to use an arbitrary constant with no justification besides "making the units check out", then that is a very bad sign.

If I say "you can measure speed by dividing force by area", and you point out that that gives you a unit of pressure rather than speed, then I can't just accuse you of nitpicking and say "well obviously you have to multiply by a constant of 1 m²s/kg". You wouldn't have to tell me why that operation isn't allowed. I would have to explain why it's justified.