Property Taxes Protest

We just purchased a house near Houston (Brazoria County) in August of last year. There was no tax increase from 2020 to 2021. The 2021 tax appraisal was $186K ($171k with the previous owner's homestead exemption), the proposed 2022 appraisal is $226k (an increase of around 21.5%), we paid $250k, and best estimate of current market value is around $260-265K. Our homestead exemption is filed, and I know that homestead caps property tax appraisal increases at 10% per year in Texas, but I don't think that will stop the current appraisal from going through at +21.5%, will it?

The house is in pretty good shape, and a new roof was put on the month before we bought it. There was some rainwater damage from a tree penetrating the roof (I removed the offending tree a few weeks ago), and you can still see the drywall patch on the ceiling where it was repaired. There is also old flooring in a good portion of the house that will need to be replaced before long. It is the laminate made to look like hardwood, and it is starting to show signs that it got wet. It's not peeling up off the floor or anything, just some barely noticeable bubbling.

I guess my point is that we could show a few minor things to reduce the value, but with us JUST buying the house, the county appraising $25k below what we paid, and property values continuing to rise, would there be any chance of us getting it reduced? I know if we can get it reduced this year it will compound with the future years being capped by homestead.