How do I argue for higher salary when they come at me with “internal equity and external competitiveness”?

Me: currently a supervisor to two departments with shy of 20 direct reports.

My org made cuts last year and they eliminated the role of a supervisor in my department. One day before I left for maternity leave, my director comes into my office and floats the idea of me absorbing those departments into my role. I verbalized an interest in helping out as their interim supervisor and that was the end of the conversation before I gave birth that weekend.

Now I’m officially back at work and my director says that this is a permanent merge to add 3 departments which I have no subject matter expertise one, and a total of 50 or so direct reports. I requested for an updated description of my job responsibilities as well as new compensation. What I received was a vague list of responsibilities and a sad excuse that I wouldn’t technically have to do much because everything can be done by the team leads (not true, I’ve seen their previous supervisor work their tailbone off). And an insulting compensation that amounts to less than a 2% increase in my salary. I reached out to the compensation department who simply stated that the new number was based on internal equity and external competitiveness.

How do I argue for more? And what would technically be fair? Is the juice worth the squeeze? If I refuse the offer, could they potentially fire me?? I might be willing to take the pay but only if it’s an interim position- how do I argue for that?

Ps. I want to stay in this organization and continue to grow as a leader - I worry that if I’m not a good leader, my failures will impact my tenure at this org. This sounds awful - I’m willing to do this job, but only for higher pay. I’m currently juggling a newborn and in grad school for my masters and I just need it to be worth my time. Please help!!