You got me going on this one so went and found an old doll my wife had from when she was a kid, similar material, which oddly had both biro and felt tip on it.
I tried about 15 products on it including some serious caustics and spirits which should of eat it.
I really see your problem, nothing worked, it seems that the biro and felt tip have etched their way into the material by absorption, with nothing to loose I started to "wet n dry" the material with 1200 and then 600 grit to see how deep it went. Although I took off a number of fine layers, it was still there.
For me there wasn’t anything I could do, however have you ever managed to get this to even fade slightly with anything?
No Im not really a doll scrubber, I just can stand being beaten by anything.