ᐅ Is it possible to install insulation between rafters without using an underlay membrane?

Hello,
I am currently planning the retrofitting of insulation for a pitched roof that was built before 1980 and so far has no insulation or underlay membrane.

Since completely removing the tiles and installing a new underlay membrane is not an option, and the method of installing underlay membranes between the rafters is too complicated and not something I fully trust, I am now looking for another solution.

During my search for the right insulation material, I came across foil-faced polyurethane boards, which are supposed to be water- and mold-resistant, meaning that technically no underlay membrane would be needed, right?

My plan would be to simply place the boards of the appropriate thickness between the rafters. However, I am not sure if an air gap is still necessary or not, so I am considering the following options:

1. Place the boards directly against the battens, leaving only the air gap between the tiles and battens, which wouldn’t be much.

2. Fix one or two battens perpendicular to the existing battens within the spaces to be insulated and place the insulation boards against those, creating an air gap equal to the thickness of the battens.

Which method would you recommend here? 1, 2, or something completely different? I am open to suggestions.

Also, I am not sure how these boards are normally fixed in place. Is it enough to simply cut them to size and press them into the space like mineral wool insulation?

If it is important, the roof is not covered with clay tiles; I don’t know exactly which type it is, I only know that they are not made of clay.