I believe glue would do it, but originally those felt channels were riveted in, before the rubber on the other side .... the channel should fit exactly, leaving a slightly beaded edge just barely proud of the channel, should be a very tight fit, might work to press something in the slot ....
this is the only pic on hand, you can just see the rivets to the bottom bracket and maybe see the woven part just above the channel - the ones on my 58 came with chrome beaded edges

ahhh, just thought - you prolly got the roll of felt strip which yes, can only be glued - the real deal is a formed channel with a metal core
Bill