To do real bi-amping, you need to replace the internal passive crossover network in your speakers with an external electronic crossover. The problem with passive bi-amping like you propose is that both amps will be presented with full bandwidth. The tube amp in particular will still be required to reproduce the power-intensive bass frequencies, which could cause it to clip, potentially destroying the tweeter.

Usually, passive bi-amping requires two identical power amps for the low and high end drivers so that there are no power limitations and the gains are identical.

Personally, I think passive bi-amping is not worth the trouble (nor is bi-wiring).

You are absolutely right about the tube amp sound quality however. An alternative perhaps would be to use two stereo tube amps, with one channel of each driving the woofers and the other channel driving the tweeters. At least this would satisfy the equal gain problem. You might still run up against power limitataions however unless the tube amps have over approximately 40 watts, or unless your speakers are very efficient. I actually use a 5 watt tube amp for my tweeters in an active electronic crossover setup, but they are extremely efficient at 106db/watt.