Wiktionary
but hey
interj. (non-gloss definition: Used to show that the thing just referred to is not ideal or does not support the argument, but that the situation cannot be improved upon, or the speaker is unwilling to try.)
Usage examples of "but hey".
Properly calibrated, it should in theory be able to pick up the radiation from the remains of Sellafield, miles to the south and west, but Hey's battery is nowhere to be found.