I am very surprised at the 90-deg to the wind that you experience. Agree, that is not going to be desirable. Yes, my boat hove to ~45-deg (or less) off the wind. And, while 40-kt that I experienced seemed pretty "brisk" to me at the time, it does not add up to the, "extreme" conditions that you originally asked about. There was relatively little interference to the boat's orientation from the sea-state (4-6-ft steep chop).
I would think 20-kt would be enough to get the boat to heave to. You of course know your boat better than I, nonetheless I suggest that you reef the main, furl the headsail completely and if the cutter sail is on a furler, reduce its size by 30%.
The idea is to get the centers of force (the two countering sails) over the center of resistance (the keel). Bring the sail force as close to the mast as you can by reefing. You may have to experiment a little with the amount of main vs cutter sail that you use to find the balance "sweet spot" on the Tayana.
With the balance right, the effect of a slight change in orientation of the hull from sea or wind gets quickly countered by "the other" (back-winded) sail. It may be that the cutaway on the forefoot of your keel moves the center of resistance far enough aft that you will need a bit more main and less headsail.
As a side note, we were surprised by the amount of leeway that even a pretty decently performing cruising boat with a relatively deep, 6'2" keel can experience when beating into 40-kt. of wind. In short, heaving to was a welcome relief!