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][1] 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.


  [1]: https://mathoverflow.net/questions/7678/free-subquotient-of-galois-representations-coming-from-hida-theory