You asked if the tabernacle is strong enough to keep the mast from swaying from side to side. Actually, none of them really are strong enough. If something is going to break I'd rather it be the tabernacle and not my mast.
My friend's Hunter 27 was sitting on the mooring one stormy night and one of his stays parted which caused the mast to go over the side. There was no damage to the mast whatsoever. The tabernacle plate which isn't a hinge plate, pulled off the cabin top where it's held with wood screws. With a little Fiberglas repair, he was back in business. These boats were designed that way for this purpose.
If you follow the advice and take it slow, your mast should come right down nice and straight with no swaying.
There are some additional things you can do to insure that this mast lowering is successful. I've seen some of the guys in the club take a long pole and Mickey Mouse a crutch to it and have a guy use it to hold and guide the mast down.
If you have a furler, it should be also guided.
The temporary boom bridles which attach to your triangular plates offer some lateral stability but it won't be enough if one of your stays gets snagged, a cross wind, or the boat leaning to one side.
I have a different set up with baby stays, pivot bridles, and a mast yoke and I still have to follow these same rules.
What I'd really like to do is try using my mast yoke with two of my baby stays on my friend Ray's O'Day 26. This yoke can be placed on any mast at arm's reach and the temporary stays (baby stays) would go from the eye bolts on the yoke to the triangular plates along with the boom bridles. Between the boom bridles and the yoke, I'm sure that you'd get a little more lateral stability.
The yoke I made up out of oak a few years ago has a hinge attached to it, but it really doesn't need to be that elaborate. It could be made up to fit snug around your mast with a long bolt to add a little more grip similar to what is seen in the last two pics of my friend Wayne's Seaward 22 mast raising/lowering Gin Pole set up.