As long as you can sync the Tyros's MIDI clock to the Tascam (or vice versa, though that's usually trickier), you can simply make multiple passes recording each track soloed in the Tyros sequencer (presumably, you are going to make an SMF of your Tyros performance and edit it before recording?) to separate tracks (or stereo tracks if stereo sounds and effects)...
And, check with a Yamaha guru for a definitive answer, but if you set up the TC to respond to any MIDI channel, set it so it sees the MIDI track of what you played on the keyboard Chord Recognition side, and it ought to track just fine during overdubs.
You ought to be able to set up the Tascam to output the recorded final vocal soloed, send that to the VH, and then record the VH voices ONLY back to the Tascam, too, so you can get the phrasing right before you take it to the VH.
Hope this helps.
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An arranger is just a tool. What matters is what you build with it..!