As usual for things where people can have strong incentives to lie, revealed preference is the only reliable indicator. But since that can't be observed in a psychology lab, instead we see questionable attempts at mind-reading like drawing spurious conclusions from response times. It would be just as scientific to hook the subject up to a polygraph and ask about biases.

Since obesity is generally seen as a moral failing rather than something that can't be controlled, people are comfortable admitting their bias. However, for race where it's rather unfair and also doing so will get you fired or worse they prefer not to.