I think I found enough space :
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The blue arrow gives 12mm. The maximum height of the battery + contacts has to be under 4mm (I tried with squares of styrene 1mm thick, with four I could no longer close the two halves)... I think a CR 1225 can fit in there, it's 12mm in diameter by 2.5mm high. The question is wether it can feed the LED. I'll try to find out soon.
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