I really have limited experience of this stuff but it might be enough to just have a few contact points around the lid face. Ideally you want something that lets you open and close the box easily.
There are various options for this. At work we have a good relationship with TC Shielding and they do things like (metal) fabric over sponge and springy beryllium 'fingerstrip' contacts. See the links below.
https://www.tcshielding.com/fingerstrips.htm
Quote:
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The robust nature of the product makes it ideal for enclosures which require a high frequency of access such as cabinets, cases and doors.
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The individual springy fingerstrips can be quite big (over 3cm long and 1cm wide) and it might be possible to fit several of them such that they sit sideways and get squished inwards as the lid faces meet.
The metal fabric over sponge strip works quite well too and you probably won't need a continuous strip of it.
https://www.tcshielding.com/fabwrap.htm
It comes in all shapes and profiles and you might only need a couple of feet of it cut into short strips.
So this could be a simple sample request from TC.