This sort of thing can get kind of complicated - especially if it has to be described verbally. Pictures would help a great deal.
To begin with, just forget about what's shown in the Heller rigging diagrams. They're notorious; the people who drew them simply didn't understand rigging. (They got better as time went on - the "Columbus ships" were among the firm's first sailing ships - but even the diagrams in Heller's last sailing ship kit, the Victory, are full of errors.
Yeah, a lateen-rigged mast normally requires a stay. It would run from a point near the top of the mast (probably above the yard - see below) to some convenient belaying point on the deck forward. There do, however, appear to have been exceptions. I just glanced through Cogs, Caravels, and Galleons, the relevant volume of the Conway's History of the Ship series; it contains quite a few contemporary pictures of ships from this period, and modern replicas of them, that don't have stays. Sometimes there's a stay on the main mast, but none on any of the others. I guess the shorter ones were thought to be sturdy enough without stays.
There does need to be a halyard to raise the yard to the masthead - and keep it there. It looks like the halyard sometimes ran over a sheave in a big block built into the masthead.
A basic feature of the lateen rig is that every time the ship comes about, the sail has to be furled, the yard has to be "topped up" (swung to a near-vertical position), and the yard and sail have to be swung around to the other side of the mast. There are various ways to do that; the following is a guess about how such a sail would be rigged in 1492. (Bear in mind that the extant information about such details is pretty scanty.)
The yard is secured to the mast by a parrel - a simple fitting that sort of resembles a roller bearing. On a small mast/yard assembly the parrel might consist of a single piece of rope running through a series of hardwood rollers (easily represented by glass beads on a model). In the case of a larger rig, there would be two or three rows of rollers, separated from each other by a set of "parrel ribs" - wood boards with holes drilled in them.
Caveat: the earliest example of a surviving parrel that I've run across came from the wreck of the Mary Rose (1545). I'm not sure this method was in use 50 years earlier, but it seems reasonable.
Anyway, the parrel has to have some sort of tackle hooked up to it that allows it to be slacked off, so the yard can move a couple of feet away from the mast. The yard has a tackle called a "topping lift," which runs from the upper end of the yard to either the masthead or, sometimes, to a point well up the next mast forward.
When the ship comes about, the sail is furled (using lines called "brails," which run from the foot of the sail to blocks on the yard, and then to the deck). One man (or maybe two or three) hauls on the topping lift, while another slacks off the parrel. The yard swings up so it's vertical (or nearly so), and two or three guys grab the lower end of it and walk it around to the other side of the mast (the lee side on the new tack). The parrel tackle is then hove taut, the topping lift is slacked off, and the ship settles down on its new course.
I had an interesting conversation about this with the gentleman who's in charge of the "Jamestown Ship" replicas (the Susan Constant, Godspeed, and Discovery). He says the crew of the Susan can accomplish that maneuver in one or two minutes. But he also says that if the ships are heading out for a day of sailing and he anticipates that there will be a lot of tacking, he won't bother to set the mizzen sail (the only one that's lateen rigged).
For this system to work, the shrouds have to be secured to the mast above the yard (i.e., the yard has to go through its various evolutions inside the shrouds). There are some pictures and models from the sixteenth and seventeenth centuries that depict the yard mounted outside the shrouds. That would make the system a good bit more complex.
All that is probably a bit confusing; like I said, it would be a lot clearer in a good diagram - which Heller didn't provide. But maybe it'll help a little. Good luck.