I've always avoided adding TT dye to water based finishes for the same reasons you found.
Mixing in with shellac (cut with alcohol) works well as a toner and sealer coat. I always spray to avoid the drips and rush of applying.
With that dry, I've had no problems with either a WB or oil based top coat.
As with BB, I hate stains and much prefer dyes, but applying to raw wood has its own issues with getting even coats as it dries very fast.