Protesting taxes when market value is set lower than your purchase price

My wife and I purchased our home in the fall of last year. Our tax appraisal this year came in about $20k less than what we paid for the home. However, the appraised value went up approximately $150k from last year because HCAD hadnt adjusted the appraisal value in like 6 years. Should I protest the appraisal even though it’s less than what we paid? If so, should I do it myself or hire someone?

Finally, will any evidence I use be in the public record and potentially hurt my home value when I go to sell down the line? Thanks!