I'd feel bad wasting the time of the other company just because my current one doesn't value me.
Also why do I want to stay with the cheap people? They proved to be unable to deliver market rates for my efforts unless I make a big scene about it. I don't like to do business with people like that.
No employers going to give you the maximum that they can afford without negotiation or a reason to. It's not being cheap, it's common business sense.
It's no different than price shopping between mechanics. You can do that while being clear on your intentions and not making any promises you can't keep