I think that an Arbor Press may be your friend instead of the scissor jack. I have the Harbor Freight and it works OK. I did some fine tuning by smoothing things up a bit but straight from the box it worked fine.
https://www.harborfreight.com/1-ton-...ress-3552.html
If you picked up some 1/2" steel plates, then filed the edges and the surfaces you could then sand the surfaces smooth right down to a mirror finish. I'd take the turntable out and lay the bottom plate directly on the flat surface then the melted plastic and top plate and then press it.
If by chance the angle of the handle isn't where you can get good pressure, the ring on the right is held in with a single set screw, you can loosen the screw remove the ring then pull the shaft with the handle out till you can move the ram up or down till you can get the right angle.
You could do that before by experimenting with how the set up would be before hand.
I'm also wondering if those polished plates could be preheated with the plastic to give you more working time :shrug: