I make leather bags for cue stics, I use contact cements for sections where the leather needs to flex.
I use Elmers glue/White Glue/PVA glue when I need the leather to be stiff.
I have seen the hole and you need to join the edges where the screwdriver made the hole, basically you would be putting glue on the inside part of the surround which is difficult when using contact cement.
I would try using contact cement but I would not let it dry and just simply bond the two surfaces while the glue is still wet
Suggestion : buy a new surround, puncture it and practice the suggestions posted by the members here
Anong brand at saan nakakabili niyan sinasabi mo flexible glue? Might come in handy even with fabrics.
Btw, have you tried using Elmer's glue? The all-purpose white glue really isn't that hard.
Have you even seen the intended application? Wala naman two pieces to join there eh, it's just a small rip in the surround that needs patching up.