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For your titular question, Beats me. Personally I'm not aware of anyone who's studied the distribution of CM points with respect to height in the way you describe.

What I have seen papers that study the distribution of CM points with respect to things other than height and papers that look at the height of CM points (as well as papers that say quite a lot about the Faltings height of a CM abelian variety).

Here are a few different papers of note on those topics:

Equidistribution of CM points:

http://www.math.ucla.edu/~wdduke/preprints/modud.pdf - Gives an amazing asymptotic result on the trace of a CM $j$-invariant (i.e. a result on $X(1)$).

http://www.math.columbia.edu/~szhang/papers/ZhangIMRN.pdf - Gives an analogue on more general modular curves and quaternionic Shimura Curves as well as a connection to the Andre-Oort conjecture

Also good for an overview is the "Equidistribution in Number Theory" volume edited by Granville and Rudnick http://www.springer.com/mathematics/numbers/book/978-1-4020-5403-7

Heights of CM points/cycles:

http://www.math.columbia.edu/~szhang/papers/HCMI.pdf - Heights of CM Points I--The Gross-Zagier formula

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.4204 - This attempts to identify affine elliptic modular curves by the presence of CM points of large enough height(logarithmic height on $\overline{\mathbf{Q}}$).

I'd like to emphasize that this does not pretend to be a complete reference list and I don't pretend to know much about the theory of heights. Rather this is a smattering of exciting ideas in the area.

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For your titular question, Beats me. Personally I'm not aware of anyone who's studied the distribution of CM points with respect to height.

What I have seen papers that study the distribution of CM points with respect to things other than height and papers that look at the height of CM points (as well as papers that say quite a lot about the Faltings height of a CM abelian variety).

Here are a few different papers of note on those topics:

Equidistribution of CM points:

http://www.math.ucla.edu/~wdduke/preprints/modud.pdf - Gives an amazing asymptotic result on the trace of a CM $j$-invariant (i.e. a result on $X(1)$).

http://www.math.columbia.edu/~szhang/papers/ZhangIMRN.pdf - Gives an analogue on more general modular curves and quaternionic Shimura Curves as well as a connection to the Andre-Oort conjecture

Also good for an overview is the "Equidistribution in Number Theory" volume edited by Granville and Rudnick http://www.springer.com/mathematics/numbers/book/978-1-4020-5403-7

Heights of CM points/cycles:

http://www.math.columbia.edu/~szhang/papers/HCMI.pdf - Heights of CM Points I--The Gross-Zagier formula

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.4204 - This attempts to identify affine elliptic modular curves by the presence of CM points of large enough height(logarithmic height on $\overline{\mathbf{Q}}$).

I'd like to emphasize that this does not pretend to be a complete reference list and I don't pretend to know much about the theory of heights. Rather this is a smattering of exciting ideas in the area.