judging by the thickness of your perspex,i agree that tapping the perspex to accepts crews would be your best bet,then you could apply self adhesive rubber draught excluder, to the box or lid,you can get it in B&Q it comes in various guises,i.e.single or double width i.e.5mm/10mm,
another way is to use the clear silicone,i.e.the one that smells of ascetic acid,but without useing the wax,smear the lid of the box with the silicone,let it cure,then screw down gently onto box when you remove it,no silicone will attach itself to the box as its already cured,
There is another way also, which i have used in the past, you will need an inside tube from a bicycle, cut the tube into strips,apply thethin rubber strips to the lid using a good quality glue, once set,drill through the rubber where the screws go,then put lid on the box,this will work every time,and the rubberwill always beattached to the lid,
Wullie