First off, based on my experience, that red dye will be....... RED! I would start with a brown dye and then, if it's not red enough, go over it with a diluted red dye. You can adjust the intensity of the dye by the amount of alcohol you dilute it with. Red birch can run from reddish brown, to brown, to cream colored, so it all depends on the wood.
With a dye stain you should not need a BLO toner, and I would not mix anything but alcohol in the dye. Haven't tried it, but I don't see mixing an oil into an alcohol base ending well.
On another note, I'm not really a fan of Truoil, simply because it isn't a "true" oil finish. It's made up of 56% paint thinner, 33% oil base varnish and 11% oil, so it's really more like a wiping varnish. If you want a true oiled finish use multiple coats of BLO or real Tung oil (yes, there's fake Tung oil), letting it dry for two or three days between coats, and finishing with 000 steel wool and wax. Use good wax, like Treewax or Minwax, not Johnson's floor wax. It has too much oil in it.
You should seriously consider using wiping varnish. It's easy to use and easy to make. Get some real, oil based varnish (like Spar varnish) and dilute it 50/50 with paint thinner. Wipe it on with a clean soft cloth and let it dry. Hit it with steel wool and coat it again. Then again and again, until you get the build you like. I will usually run some 600 grit lightly over it after four coats, just to knock down any debris and even the surface.
Hope this helps!
