Take a look at section 1 of M. Saito, Arithmetic mixed sheaves, Inventiones 2001. Part of the data of his category $MHM(X/K)$ is a bifiltered $D$-module defined over $K$. So it seems your question has a positive answer. But as always with this stuff, I might be overlooking something. If it's really important to you, you should ask him directly  (he's not on Mathoverflow as far as I know).

Take a look at section 1 of M. Saito, Arithmetic mixed sheaves, Inventiones 2001. Part of the data for an object in his category $MHM(X/K)$ is a bifiltered $D$-module defined over $K$. So it seems that your question has a positive answer. But as always with this stuff, it is easy to overlook something. If it's really important to you, you should ask Saito directly  (he's not on Mathoverflow as far as I know). Alternatively, as Matthew Emerton has suggested, it may be simpler to work out the cases you need by hand.