The assumptions on the SI units are not correct.
The kilogram isn't 'defined by an arbitrary weight', it's defined using Planck's constant since 2018:
The kilogram is defined by setting the Planck constant h exactly to 6.62607015×10−34 J⋅s (J = kg⋅m2⋅s−2), given the definitions of the metre and the second
Hence, the kilogram is now essentially defined in terms of the second and the metre. Though it did used to be defined by a platinum-iridium cylinder (and before that a pure platinum cylinder), whose mass was (as close as possible) to the absolute weight of a volume of pure water equal to the cube of the hundredth part of the metre, at 4 Celsius.
The metre is not a derived unit in SI, it's one of the base units, defined in 1983 as:
The metre is the length of the path travelled by light in vacuum during a time interval of 1/299792458 of a second.
I'm not too sure where the water based definition you mention comes from, but if you can elaborate, I can see if I can see where it fits into the story for you.
You are correct that the Joule is based on the kilogram, metre and second (since $J = kg \cdot m^2/s^2$). And if you did a dimensional analysis on the Joule, you would find it based on the above combination of the units you mention, it doesn't necessarily mean whatever energy you are measuring (in this case some particle, i presume) is correlated to that cylinder of platinum or any other thing used to make a definition.
Put simply, a unit is just shorthand for saying "is equal to this many of a quantity". For example we can define our own unit, the Dave, as equal to 63 times the average mass of a male platypus. I weigh 1 Dave (yeh, I need to lose some mass...). It doesn't mean that me and the average male platypus are in anyway correlated.
To answer, the 'extra constant' bit, what I think you're asking is:
Why is there no constant of proportionality used in $E=mc^2$ to relate the Joule to kilogram, metre, and second, since their definitions seem unrelated?
Feel free to correct me if I misunderstood. Constants of proportionality (and other constants) are used, as a way to get a certain unit out of an equation. $E=mc^2$ is just an equation, it doesn't require any constant until you choose a system of units to work in. Let's try adding the constant you mentioned, we will call it, k, so now the equation is $E=kmc^2$. For getting an answer in Joules, from kilograms, metres and seconds, k would be 1 (it is how the Joule is defined). If you wanted an answer in milliJoules ($1/1000$ Joules) from kilograms, metres and seconds, k would now have to be 1000, to balance the units. If you wanted an answer in milliJoules, from grams ($1/1000$ Kilograms), metres and seconds, k would be back to being 1, since the alterations made to both sides balance.
A good real world example of this, is natural units, where c=1 (whereas, in SI it's 299792458 $m/s^2$, but here the value and units are adjusted to output an energy in electronVolts, instead of Joules), and Energy and mass are measured in eV, but the equation is still $E=mc^2$. All the equation says, regardless of any constants, is it takes this much energy to create this much mass from the vacuum.
Hope that helps, feel free to ask more if I was unclear or didn't cover some of your questions - either units or platypus related.
SI Unit Definitions