If I read your question correctly, I think you broke the thermostat for the auto trans cooler. Easily done, and common problem. I can't remember, but if you can put it back together without leaking coolant, it will be fine to drive. Then just replace the thermostat insert when you get a new one.

Without that piece, your trans fluid temp will be slower to come up to temp, but that's not a big deal unless you live in a very, very cold climate.

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2012 xDrive35d
2001 BMW X5 3.0i 5MT
and two 1987 Corvettes - Callaway Twin Turbo and Guldstrand Grand Sport 80
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