Ah, now I see. I was thinking of a walk-thru transom - like a small door on the port or starboard side.
Yeah,I know what you mean about those slides. The transom door (walk-thru) on my Sundancer does the same thing (but only when I'm trailering). The only plus is I installed magnets in the door and the side of the boat to hold it open. Luckily, they're strong enough that if it does open up during trailering, the magnets hold it and keep it from slamming back and forth.
But, just for reference, if you did want those slides to stay put... Install a short section of black, nylon strapping just below the "knob" when it's in the closed position. Install a snap just above the knob and a snap in the other end of the strap. Snap the strap over the knob, erring on the inboard side of the knob.
Another way to go about it is to make the a channel that the outboard edges of the door slide into, from top to bottom. Maybe just an extended piece of flat stock on the cockpit side. I assume there's chocks on the engine side? I've never seen one of these doors in person.
Sound like you've got a nice laundry list of things to do - the Gulfstream is a really nice boat. I guess all you'd need to do is add a pad to the drop-door (after securing it better) and a removable bench (as simple as wood) - which could be held to the transom wall with quick release-type latches. If I understand you correctly, you've already got a bead on having something made. If not, maybe we can hash out an idea.