Is there a system of units that replaces time with light-distance? I would like to start this by saying that my motivation for asking is that I find relativity very difficult to deal with using the SI system. It strikes me that the problem with this system is the fact that it was based on an earlier system that assumed time to move at a constant rate under all circumstances. Since we now know this not to be true, is there, or should there be a better system out there? One that is based on the constant speed of light, not a constant rate of time.
For example, if MKS became MKL where 1L = 1 Light-Gigametre (the time it takes light to travel 1 Gigametre - not in seconds) then in the world of slow moving 
objects we would have 1s approximately equal to 1L/3 and all would be fine. 
By my reckoning (and probable naivety) 
it should also hold out in the world of fast moving objects where time starts misbehaving but L remains constant.
Does such a system already exist?
UPDATE -
What I am looking for is definitely not some scaled version of our existing base units. 
I'm looking for a system of units that ditches time altogether and starts from scratch with light-distance in its place.
A fundamental change that affects all of the units that have a time component.
I was prompted to look at Planck Units and in partiular Planck Time. At first sight this appeared to be what I was looking for 
i.e. it is based on light-distance. Unfortunately it looks like this has just been equated to a fixed number of seconds. 
The problem is that as soon as you make it a scaled version of time you have lost its true meaning and it is no longer true 
light-distance. It can't be, because light travels at a constant rate and time doesn't.
Update 2
From a practical point of view, as a stationary observer on Earth, during the same period of our clock based time I could be looking 
at a light bouncing off a distant stationary mirror and measure one distance of light (2d) or I could be watching the same scenario on a fast moving passing 
vehicle and measuring a different distance of light (a V shape of height d). So in effect, the light-distance measured by me (L) would reflect the amount 
of earth time being experienced at the source of the light. So L (unlike t) would be dependent on what was being observed, as it should be. Or should it be that in a system where light-distance is replacing time we should just say that in both cases the light moved by 2d?
 A: Based on the discussion I had with @Alan Gee in the comments on the question and later in a discussion, I will give my answer to that question.
A unit system defines units of measurement as definite magnitude of a quantity. A unit system does not make any statements about the constancy of time or space or anything really. It defines a refernce against one can measure quantities nothing more nothing less. It has nothing to with physics and makes no statements about it.
The laws of special relativity (SRT) have nothing to do with any units. We can calculate with SI units, with natural units with Planck units: with every unit system we please and we will allways get the same result. The constant factors in our formulas may change and get easier or harder but the physics those formulas encode does not depend on a unit system.
There is and never will be a unit system that can solve the "problem" that SRT has some unintutive and mathematically tricky points. Maybe one day there will be a more elegant theory of relativity but this theory will be based on a different mathematical description of physics and not on the units we calculate in.
To the second part of your question: The SI units fixed one unit with the speed of light: the meter is defined as the distance light travells in 1/299792458 seconds. So the meter is fixed by the speed of light. You can not use the speed of light to fix another unit. Anyway it does not make a difference for SRT how you fix your units. Even the totally arbitrary old SI system (with Prototype Metre) had no problem what so ever with SRT and SRT had no problem with SI.
A: Two points: First you say "It strikes me that the problem with this system is the fact that it was based on an earlier system that assumed time to move at a constant rate under all circumstances." Distances are also not absolute in SR. Second: I think this is up to you, for example when travelling from A to B it is conventional to use the distance between them as X km. However, it is often more meaningful to say that the journey takes Y hours (where the speed is taken as the maximum as allowed by the limit on the road used). a more physics based example would be: Alpha Centauri is often quoted as being 4.37 light-years away, so that you could argue that "Alpha Centauri  is 4.37 years away" with the understanding that the "maximum allowed" speed is the speed of light.
For example consider the Lorentz transformation (just 2D):
$$t' = \gamma (t - \frac{vx}{c^2}), x' = \gamma (x - vt)$$ If you wish you can write in the form (with $\beta = \frac{v}{c}$) $$ct' = \gamma (ct - \beta x), x' = \gamma (x - \beta ct)$$
or as $$t' = \gamma (t - \beta \frac{x}{c}), \frac{x'}{c} = \gamma (\frac{x}{c} - \beta t)$$ In the first form the units on both sides of the equations are in $m$, while in the second form the units of both sides are in $s$.
As well, in special relativity it is possible to have (for non-zero mass particles) space-like events or time-like events.
A: I will attempt to answer my own question, award some points to the best answer and close this off.
It looks like the answer is No, there is not such a unit system.
Should there be? 
Well my own version of the classic SR example has shown that although it may make SR calculations neater and deal with time-dilation (by removing time),
working with light-distance instead of time doesn't negate the effects of space-contraction (a point that I think jim was trying to emphasize) and still leaves us with a situation where light-distance is dependent on frames of reference. So probably overall it is not worth the effort.
