Mosuli space of SL_2 bundles on any genus 2 curve - is P^3 by Narashimhan and Ramanan. So working with P^3 should be accessible, but may be not so easy as might seems...
Classical Hitchin system has been described by van Geemen, Previato, Gawedzki, et. al.
Quantum hamiltonians should be contained in Geemen de Jong paper: http://arxiv.org/abs/alg-geom/9701007
So one needs to 1) Quantize (?may be done) it 2) Make Hecke transform and see that result is as predicted by Beilison and Drinfeld - product of initial Hitchin D-module on D-module give by SL_2-oper.
If I would think on this I would try to do the following:
1) find Lax matrix L(z) - it might be done by Gawedzki, et.al.
2) try to guess what is the "qauntum spectral curve" "det"(d/dz - L(z)) - it gives both quantum Hitchin hamiltonians and SL_2-oper (this done by Talalaev for Gaudin-Hitchin for P^1 http://arxiv.org/abs/hep-th/0404153 )
3) try to prove that Hecke transformation agrees with "qauntum spectal curve"
As in previous post I would strongly suggest to use down-to-earth point of view on Hecke transformation.
So in such a way one may prove that the Hitchin's D-module is Hecke eigen-sheave with "eigenvalue" given by the SL_2-oper="quantum spectral curve" (D-module on basic curve).
Considerations of other Hecke eigensheaves (not Hitchin's) is another story...
In our paper http://arxiv.org/abs/hep-th/0303069 we described classical Hitchin system for the degenerate genus 2 curve y^2 = (x-a)^3 (x-b)^3. However we did not check that our "analogs" of Narasimhan-Ramanan coordinates are indeed limits of true Narashimhan-Ramanan coordinates, so it might not be very helpful. Also Talalaev's formula cannot be applied directly for the Lax matrix in these coordinates. It is coordinate dependent, it is solvable problem but requires some work.