I take a similar approach. It’s fine (and good business practice) for buyers to look around to find the right seller to work with. I assume that most sensible buyers do this. It’s important.
However, once a buyer has approached me, if I get a hint that they are trying to play me off against another seller - then I’m not interested in working with them.
That approach tells me so much about their values and morals (or lack of). A common approach is “X seller can do the job X dollars cheaper, but we want to work with you. Can you match their price?” My response is “Then work with them”.
Very occasionally I will go further and put them in their place by saying “You have demonstrated a lack of respect for my skills, experience and time - and therefore I’m not interested in working with you. Go with the cheaper seller”. It might not be an entirely professional response, but it makes me feel better!