Well, the answer is: the synthetic spider silk does take dye.
It takes dye very well, pretty much identically to how the bombyx does. It has a slightly more tan shade, but not as dark as tussah silk.
This is the spidrex (right, small skein of twisted singles) next to bombyx (left, large skein of organzine doubles) that went through the same dye vat. The spidrex is slightly darker, but not much.
This shows a few different colors. Left to right, they are: a blend of violet with hot fuchsia, royal blue, a blend of sun yellow and aztec gold, and cherry red. It seems to take the various colors well. There was a little bit of bleeding with the yellow, but I had a very saturated bath. The blue surprised me – I thought it was going to be too dark, but it has a lovely sheen. All the dyes are Jacquard acid dyes.
The technique I followed is the same as what I use on regular bombyx silk: wet out in warm water with citric acid and a couple drops of calsolene, then place into the hot dye solution (also with citric acid and calsolene) over low heat, simmer 20 – 30 minutes without ever boiling (although the yellow vat did boil briefly on accident, and appears to still be fine) and then rinse out in warm water, then a final rinse in Milsoft.