If the competition is that high, the farmers will have very low profit margins. Any extra expense needs to be added to the price.

This won't affect competition much, as all farmers have the same tax and therefore the same need for increased price.

The only noteworthy thing I see here (based on your information) is that the extra tax is per cow, but the price increase will be on the milk. This will hit farmers with lower amount of milk per cow. I guess this is farmers with more focus on quality and animal welfare.