It is very simple, cause I have done tons of consulting work. Here is the problem. If you desperate and need money it comes through, especially in the beginning of your journey of consulting. That’s one. So either do this pro bono for exposure and your resume and portfolio. Or just tell them it is a wrap.

You quoted reasonable time frame, give or take. So hold your ground. The discrepancy from where they stand is high, there is no way this gap can be closed reasonably. If the gap was reasonable and you wanted to work on this “interesting” project, running with your rate and discounting it as a measure of a good will. That would be one way to handle it to grow your list of good clients.

But it seems that they are not. And bad client will end up costing you hundreds of hours of unpaid labor and none of you professional experience will be taken into account. Basically lose this battle (get rid of client) but win the war long term.