A knock on the door and proper identification are really all the courtesy that is needed. It sounds like your PD handled the situation very professionally. This story is how it used to work everywhere. Where I grew up we had a lot of woodland searches, complete with helicopters with searchlights. The police were always 100% respectful of property owners and they never caused any damage. Heck, they’d even secure the fence gates behind them so the livestock would stay put. I can imagine what one of those situations would be like if it went down here today. It would look like a war, and there would be fences down and fields torn up. That would be all, if we were lucky.
Bingo! Stop and ask/inform. This notion of entering onto someone’s property in hot pursuit needs to be tempered with care and consideration. To enter without permission and shoot the pet—the LEO’d have more than the perp to worry about. That is BS!!