What is the number n of points in your matroid?
If n is at most 9, you can simply run by the ~400.000 matroids on at most 9 points. The sage matroid package will enumerate these matroids in half an hour. There is also a database of matroids that goes a bit further here: http://www-imai.is.s.u-tokyo.ac.jp/~ymatsu/matroid/index.html
I may not fully understand your original problem, but perhaps this is useful: given the 3-term grassmann-plucker relations and the fact that certain $k$-sets are dependent, many purely multiplicative relations follow. E.g. for $k=2$ and on a 4-set $\{a,b,c,d\}$, there is a g-p relation $[ab][cd]+[ad][bc]+[ac][db]=0$. If $ab$ is not a basis, then $[ab]=0$ and hence $[ad][bc]=-[ac][db]$. The multiplicative group generated by the nonzero brackets [..] and these relations is called the Tutte group in case of a matroid. This group is a lot easier to handle computationally that the ideal generated by the g-p relations. Perhaps creating this group will help you to make your guess for the right answer. If I understand your problem correctly, this group is a sub-ideal of the ideal you are looking for.