I haven't used alkanet enough to have a true grasp of its possibilities.
I have created nice pinks with French pink clay and pale blues with woad and indigo. I bet you could do all three in different combinations to get some very nice purples.
Ultramarine violet. I gave up looking for a 'natural' lavender. Anyway I get a lovely lavender with it. Got mine at TKB Trading but I believe others probably carry it. BTW- I have the same problem with alkanet.
Rhubarb root infused in oil makes a lovely pink color. Maybe if you'd combine it with some alkanet infused oil, it would be more purple? Not sure though, I think I'll give it a try myself.