This is not always possible.
Suppose we could always find such an $M$. Take $\bar{\rho}$ reducible at $p$ with scalar image of the Frobenius. Then $M\otimes_{R_{\bar{\rho}}}R^{\operatorname{ord}}_{\bar{\rho}}$ is the universal deformation with the ordinary condition at $p$ and also occurs as a quotient in the cohomology of a (tower of) modular curves (because the ordinary Hecke algebra is just the image of the Hecke algebra under the ordinary projection) and furthermore one could descend to a fixed weight and level in the same way. So there exists a lattice in the cohomology of modular curves with comparable properties for forms of given weight and level. But this is known not to be true, as explained in the answer to this old question of mine, which is also the one considered by MO to be the most related to yours (an indication that MO does a good job).
Above is a specific counterexample which you might want to discard (for instance by taking $p\neq 2$) but proving a positive result in the direction you wish (assuming suitable supplementary hypotheses) seems to me to be very hard.
