Is it possible to derive the Michaelis-Menten equation under conditions where the product formation is reversible
No it is not possible to derive the equation for the reversible case. This is because the Michaelis-Menten equation is derived on the explicit assumption that the substrate concentration is much greater than the product concentration so that the back reaction does not occur. This is true at the start of a reaction in the lab when you add enzyme to your substrate. At later times when product builds up these conditions no longer hold and the M-M equation breaks down.

It is possible to derive a different equation for the reversible case (as some other posters do) but this begs the question of the importance of the Michaelis-Menten equation for biology. The poster seems to think that it is some universal truth, important in its own right or because it models the reaction in the living cell. I think this misses the point, which I think is as follows.


It originally was a prediction from, and hence a test of, the hypothesis that enzymes have a single active site at which the substrate binds and is converted to products.
It allowed two fundamental aspects of this enzyme model to be quantified: (i) the binding affinity of the substrate for the active site, as the inverse Km, and (ii) the catalytic efficiency ('smartness') as the turnover number or kcat, which is derived from Vmax (and enzyme concentration).
These enzyme constants can be used to address biological questions, for example whether an enzyme capable of a particular reaction in the test tube is able to account for the observed reaction rates in a tissue. Or whether the substrate you find the enzyme can use is likely to be the physiological substrate in a tissue.
It turns up enzymes that do not exhibit M-M kinetics, implying that some aspect of the model does not apply to them. This can be of mainly chemical interest (ping-pong kinetics and the like) or of physiological importance, as in allosteric kinetics (S-shaped rather than hyperbolic curve) shown by multi-subunit enzymes where the individual subunits show co-operativity, allowing more finely tuned regulation. 
And if you do want to model reactions within cells, the kinetic constants can be of use. However as one is normally dealing with a chain of reactions in an open system the mathematics are complex (and nothing to do with the equation for a single reversible reaction in the test tube).


Most biochemistry students (and many teachers) would be hard pressed to explain why they should learn the derivation of the Michaelis-Menten equation. The answer is partly so they see that it is based on a particular model of the enzyme, but also so that they are aware of the assumptions on which it is based. Hence they need to be aware that the Michaelis-Menten equation is only true for initial rates of reaction, otherwise their determination of the practically and biologically useful Km and Vmax will be incorrect.