... try letting them know your position tactfully, and be reasonable by trying to understand your own demands in context, with eye on both industry norms and norms within the company, and creating a good outcome for both sides.
And dispense with this 'integrity' moralizing.
Much like young CEO's trying to nego with VC's, hiring managers and companies do this day in and day out, they see it all, moreover, they have a different set of objectives, which you have to respect and build them into your consideration.
It's definitely a good idea that they know you might have competing offers, or that you are strongly considering moving on, but those are things you definitely don't want to browbeat them over. A simple, tactful acknowledgement is good enough. I would recommend having a quick chat in-person about it. And stay away from hard lines or antics.
The grass is always greener on the other side, wherever you go it'll be great for 6 months and then you'll discover a bunch of things you don't like ... while there's definitely a lot to be said for moving on, sometimes I think that grinding is definitely under appreciated.
people evaulate what they are doing constnatly
Employer might not have even known quitting was on the table if there is only a hint. Be bold and direct to reduce miscommunication and you'll be the one standing with integrity.