Word for dead or dormant, but not definitively one or the other?

Apart from motionless (as already mentioned in another answer), still , immobile & lifeless work here. Especially, lifeless has the ambiguity you desire. Stationary can be used too but I recall it being used more for inanimate objects.

"I carefully approached the lifeless/still/immobile mouse in the corner."

ODO:

lifeless ADJECTIVE

1 Dead or apparently dead.

‘When firefighters pulled his lifeless body from the river, his heart wasn't beating and paramedics feared the worst as he was taken to hospital unconscious.’

still ADJECTIVE

1 Not moving or making a sound.
‘the still body of the young man’

immobile ADJECTIVE

1 Not moving; motionless.

‘The body is so mute, so immobile that one might think she is dead.’