For PC version to climb the net: Standing next to the net, face the net (look at blue flashing spots). DOUBLE JUMP (2x SPACE) and the rafter immediately press G. The rat will hold on to the net. After that you can use the normal move buttons to move. Worked for me!
For PC: Face the middle of the net (twinkling blue spots) while standing close to it. DOUBLE JUMP (2x SPACE) and immediately thereafter press G. The rat will stay on the net. Worked for me repeatedly.
PC version: Face the net. DOUBLE JUMP (2x space) and immediately press G. Worked for me.