FIR filter digital differentiator with low cutoff
What you want is what I call a Jekyl and Hyde design. The low pass is parsimoniously represented in terms of cosines and the derivative in sines. So let Jekyl be Jekyl and Hyde be Hyde.

The way to go very low would be a multirate approach.  Low pass and decimate for a few cascades until the high pass part of your differentiator is a reasonable fraction of the band.  If you need the original sample rate, upsample appropriately.  Your filter plot implies that you really aren't interested in most of your original band, so why retain it?