Good to see you back on the Forum!
Few options:
1. Use a normal frame with a deep enough rebate to cover a good amount of the flat edging of the map, which should also cover the hole made by the previous hanging method. The map will be proud of the front face of the frame though, so the customer may not like that.
2. Use a deep frame and set the map back using spacers, which could be painted to match the frame itself if needed. This may cover the hole made previously. You could glaze this option, but most people don't want this kind of map glazed.
With both methods you could tape the map around all edges to a backing board which will keep it in position.
3. You could use a tray frame, using something like self adhesive foam tape to hold the map in position, but that won't cover the hole made previously. That would be the most vulnerable version of framing it.
It is likely to be the appearance the customer is going for that will drive which option you choose.
One final thought on possibly disguising the hole - mix up some acrylic paint to the same colour as the surrounding plastic and then fill the hole in layers, let each one dry before adding the next, until the hole has been filled and is far less visible.
