I found Jute at a local material shop called "Foam and Fabric". They sell material for making clothing as their main item but also have upholstery material for furniture, curtains etc and sell foam too. Bought all the internal stuff needed to re-work my MGA seats. The Jute is used in the back of the seats on those cars - convertibles so no headliner. I'm glad my Jute was still in good shape on the Tbird.
I use the foil stuff on the MGA floorboards - and they are really boards - not metal. The center console is metal but covered with the foil.
This is similar to the upholstery stapler I borrowed - however his was older and smaller than the electric ones you can buy at the home centers etc.
Here's a pic with a "hand" so you can compare the size.
I agree - don't think the poly alone would be good enough.
I wasted a lot of time with a new electric stapler.
Might try one of the cheap air powered ones from Harbor Freight? although looking it doesn't seem that HF has short enough staples.
Oh and cut the light hole small and last after steaming all the wrinkles out. The hole I cut in mine was fine until I steamed the headliner. The hole pulled very close to the front edge of the light. Thought I was going to have to add a little material but it worked - just barely.