If your basses are significantly different in terms of output, then you'll need to level them some how. You could just do it with the volume pot, but for whatever reason people don't like doing that. The Boss LS-2 can have two different volumes set which you switch between. I'm sure there are other similar products.
You probably want to make sure the volume's about the same right at the start of your chain, before pedals. Then, get the engineer to take his DI after your effects. I've never heard of a sound man refusing to use someone's effects before!! I've always had them put the DI box ontop of my amp, and then ask for the cable that I normally plug into my amp (which in your case is after pedals).