Typically, people who have a higher-than-average concern for the welfare of animals also have a higher-than-average concern for welfare of people. Compassion and empathy seems to grow for both, not be a trade-off.
Compare an analogous concern to the one you raise: "what if caring about people of color comes at the cost of empathy for your fellow whites?"
Compare an analogous concern to the one you raise: "what if caring about people of color comes at the cost of empathy for your fellow whites?"