What glue to use?

http://www.thistothat.com/

If it's OK for the glue to show, we recommend:

<dl><dt>3M 80</dt></dl>

But if you don't want the glue to show use:

<dl><dt>3M 77</dt></dl>

this is their advice I would use polyurethane hot glue and use a  wood roller to flatten the glue.

 

Franklin International HiPURformer 1361 Applicator Tool

 

I have used this on foam, wood, styrofoam and plastic it is fantastic, I have built entire structures of wood using this with plastic windows and doors, they are rock solid.

UHU POR STYROFOAM ADHESIVE40G 2
UHU POR STYROFOAM ADHESIVE40G
Can get onAmazon.

I ended up using 2 sided carpet tape and thus far it has worked we'll  It is much easier to use than contact cement.