A flat steel plate (of appropriate size) screwed into the box end with any needed box-interior support provides the needed anchor point for strings. 1/8" holes drilled for strings (at the appropriate spacing) saves real estate between the box--end and the bridge.
This should give you the anchor point you need and the spacing from bridge. I use something like that on guitars I want to be as compact as possible.
Making your own plate gives you a better sense of satisfaction, but to save time I use cheap hardtail bridges from China (via eBay), remove the bridge saddles and use the L shaped bracket you are left with. I add extra wood in the box to screw into and you are sorted. Pricewise a 6 string hardtail is under £3.00 and those for 3 or 4 strings have dropped in price too for example https://www.ebay.co.uk/itm/1X-Cigar-Box-Guitar-Parts-3-string-Hard-...
Although cheap the trade off is the wait for delivery (currently about 2 weeks to the UK, but can be a month or more) so may be too long a wait this time, but for future builds it might be worth thinking about. CB Gitty stocks 3 and 4 string hardtail bridges but they look better quality and cost a bit more - this would give you a much quicker delivery.
Here are a couple of six string box guitars I used this method on - one with the L pointing into the body and one away.
I use piano hinge attached right to the box end.
So I found a decent hinge, this box has a large lip all around but will be screwed down to box, knowing this is that enough strength if the hinge is secured to the side of the box lid itself? Box lid is about 1/2" overhang and 1/4" thick