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Dre comments on Estimating the kolmogorov complexity of the known laws of physics? - Less Wrong
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First not: I'm not disagreeing with you so much as just giving more information.
This might buy you a few bits (and lots of high energy physics is done this way, with powers of electronvolts the only units here). But there will still be free variables that need to be set. Wikipedia claims (with a citation to this John Baez post) that there are 26 fundamental dimensionless physical constants. These, as far as we know right now, have to be hard coded in somewhere, maybe in units, maybe in equations, but somewhere.
As a reference for anyone encountering this discussion, I thought I'd mention Natural Units explicitly. Basically, they are the systems of units that particle physicists use. They are attempts to normalize out as many fundamental constants as possible, exactly as you discuss.
Unfortunately, you can't build a system that gets them all. You are always left with some fraction over pi, or the square root of the fine-structure constant, or something.