[morzh]

I have succeded (almost 100%) with fixing this.
It would be a 100% but while trying to push a piece of saran wrap under the cloth and frame while gluing things (and I was being very careful) I ripped a little one place, very close to the edge of the frame. This cloth is obviously fragile.
I only found it today when unclamping the piece and seeing the wrap protruded through the rip.

I will use fabric glue and paste a piece of another speaker fabric over the rip to stop it from spreading.

Otherwise (will post the photos later tonight) here's what I did:

I split one of the paint stirrers in two longitude-wise, the length happened to be exactly what I need, and using a chisel, carved out some wood in the middle of each one so when they are crossed they'd be flush to each others surfaces.
Then I glued them together.
Then I took the old pieces of cardboard from the original frame, trimmed them a bit and glued on top of the four cross beams.
This is to compensate for the fact that the cross will be resting on top of the old frame and so it will not contact the grill fabric, and so will not support it. With the pieces in place they will contact the fabric exactly where they used to.

This solved the problem.