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This might not be what you are looking for, as they use the actual full ring of differential operators (in Berthelot's theory, your "full ring" would be $D^{(0)}$, if I understand correctly), but the following papers are very beautiful in my opinion:

Gieseker, D. - Flat vector bundles and the fundamental group in non-zero characteristics.

dos Santos, João Pedro Pinto - Fundamental group schemes for stratified sheaves. J. Algebra 317 (2007), no. 2, 691--713.

Hélène Esnault, Vikram Mehta - Simply connected projective manifolds in characteristic $p>0$ have no nontrivial stratified bundles

You'll of course notice very quickly that in all cases, the D-Module flavor is lost, as a $O_X$-coherent D-module can be translated into the world of vectorbundles vector bundles thanks to Frobenius descent.

1

This might not be what you are looking for, as they use the actual full ring of differential operators (in Berthelot's theory, your "full ring" would be $D^{(0)}$, if I understand correctly), but the following papers are very beautiful in my opinion:

Gieseker, D. - Flat vector bundles and the fundamental group in non-zero characteristics.

dos Santos, João Pedro Pinto - Fundamental group schemes for stratified sheaves. J. Algebra 317 (2007), no. 2, 691--713.

Hélène Esnault, Vikram Mehta - Simply connected projective manifolds in characteristic $p>0$ have no nontrivial stratified bundles

You'll of course notice very quickly that in all cases, the D-Module flavor is lost, as a $O_X$-coherent D-module can be translated into the world of vectorbundles thanks to Frobenius descent.