The QM probability density as a function of the wave function is given by Born's rule as a postulate. This leads to the probability density being invariant to the phase of the wave function.
Suppose instead that I leave out Born's rule, and instead more generally postulate that there exists a probability density as a function of the wave function, but invariant to the phase of the wave-function. How does this restrict the form of this probability density function?
If I understood correctly, you want to postulate a more general Born rule while continuing to require phase invariance. As a matter of fact, this won't give you any advantage due to Gleason's theorem
Theorem. Suppose $H$ is a separable Hilbert space. A measure on $H$ is a function $f$ that assigns a nonnegative real number to each closed subspace of $H$ in such a way that, if $\{ A_i \}$ is a countable collection of mutually orthogonal subspaces of $H$, and the closed linear span of this collection is B, then $f(B)=\sum_i f(A_i)$. If the Hilbert space $H$ has dimension at least three, then every measure $f$ can be written in the form $f(A)=\mathrm{Tr}(WP_A)$, where $W$ is a positive semidefinite trace class operator and $P_A$ is the orthogonal projection onto $A$.
In other words, in dimension $d>2$, any probability measure on a Hilbert space is generated by a quantum state following the Born rule, so even if you came up with a different measure it would be possible to express it as the Born rule.
The Born rule is a statement of the conserved nonrelativistic Noether charge distribution, divided by e. I don't consider it a postulate. Since as far we know the electron is a point charge and its charge is immutable, the charge distribution tells us where the electron is. You may consider other conserved quantities such as energy-momentum but these obviously do not tell us where the electron is.
