Hmm, I can't seem to figure out how to apply your postulate 4 to reproduce the phenomenology of entanglement. Perhaps we can work through a concrete example together?
First, picture two two-level quantum system in your favourite physical realization---trapped ions, electron spins, polarized photons, whatever. I'll write as if I had electron spins in my mind, but of course everything readily translates. We can measure the electron spin in two bases,
[math]\lbrace|0\rangle,|1\rangle\rbrace[/math]
and
[math]\lbrace|\rangle=\frac{1}{\sqrt{2}}(|0\rangle|1\rangle),|-\rangle=\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)\rbrace[/math]
Now, we let the two particles interact to produce the state (in conventional quantum mechanics)
[math]1: \frac{1}{\sqrt{2}}(|0\rangle_A|0\rangle_B |1\rangle_A|1\rangle_B)[/math]
Here, the index refers to the party that has access to each particle, A for Alice and B for Bob. Now if Alice does a measurement in the {|0>,|1>} basis, it's plain to see that she will either obtain the 1 or 0 with 50% probability each. If Bob then does a subsequent measurement, he will obtain the same value as Alice with certainty---their measurements are perfectly correlated.
However, in your case, according to your postulate 3, after the interaction between both parties, we have either the state
[math]2a:|0\rangle_A|0\rangle_B[/math]
or the state
[math]2b:|1\rangle_A|1\rangle_B[/math]
with presumably 50% probability for the alternatives (though you don't seem to discuss how to get the usual Born probabilities in your framework). Now, if Alice does a measurement, she will again get either 0 or 1 with 50% probability, and again a subsequent measurement by Bob will agree with certainty. So all is well so far.
But now consider measurements in the {|>,|->}-basis. In this basis, the state 1 is written as:
[math]3: \frac{1}{\sqrt{2}}(|\rangle_A|\rangle_B |-\rangle_A|-\rangle_B)[/math]
So the same conclusions as above apply: Alice measures, gets with 50% probability either or -, and Bob's subsequent measurement will perfectly agree.
But in your case, the state (for example) 2b expanded in the {|>,|->}-base, is:
[math]4: \frac{1}{2}(|\rangle_A|\rangle_B - |\rangle_A|-\rangle_B - |-\rangle_A|\rangle_B |-\rangle_A|-\rangle_B)[/math]
So if Alice now measures in this basis, she will obtain either or - with 50% probability, but if Bob then performs a measurement, he too will obtain either option with 50% probability, independently of what Alice has obtained! So their measurements will be wholly uncorrelated, in contrast to the standard quantum account on which they will be perfectly correlated.
So, my question now is, how does your postulate 4 work to avert this disaster?