A GPS unit would normally get its time from the satellites...
No idea what the batteries are - but typically this sort of thing is a Lithium button battery or maybe something a bit more unusual like a 1/2AA. I'd wait for Icom's response and if they say they can't do anything for you I'd say you have nothing lost to open it up and see what's inside. Post a pic with a ruler in it for size and any markings and someone on 'ere is bound to know...
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