What is the colour change that would be observed if carbon dioxide gas were bubbled into a mixture of calcium oxide, bromothymol blue, and water?

Bromothymol blue is acid base indicator which has pH intervals 6 - 7.6 and its two colors are yellow in acid and blue in basic.

The pH of CaO is about 11.5 to 12.5 this will make the indicator has blue color at the beginning.

By bubbling CO₂ to CaO this leads to formation of CaCO₃ which also basic (pH about 9.5 - 10) so the color of indicator remains blue

No color change of the indicator will observed (remains blue)