This is a sort of negative-leaning answer to the question about existence of software for your purpose. There is quite a bit of history to the problem in prime characteristic, going back to isolated examples found in the late 1970s by Mumford and a student of his which showed that the classical ideas could break down. The rank 2 example $G_2$ lends itself to picture drawing and has been looked at in considerable detail. See the recent updated preprint by Andersen and Kaneda here.
Andersen's clever sheaf cohomology techniques (exploiting the Frobenius map) combined with my more speculative predictions tend to imply that the results depend heavily on Kazhdan-Lusztig theory for the affine Weyl group (of Langlands dual type). Moreover, the non-vanishing of cohomology seems to involve the actual module structure, so dimensions appear only as a byproduct of the study of generic module filtrations crossing Weyl chamber walls. The algebraic group of type $G_2$ already indicates how systematic but complicated the results will be in general, so any computational approach must take this case into account. (The results for $A_2$ and $B_2$ worked out by Andersen following his 1977 MIT thesis work are also subtle, but con't compete with the complexity of $G_2$ whose alcove geometry is richer.)