I'd also suggest talking to the recruiter who floated your name. They might be able to give you good advice. But be careful if they work for the company or were retained by the company.
Might as well ask a car salesman if you should buy a new car, or a real estate agent if you should buy a house. Since they only get paid if you go through with the deal, and only get an incremental benefit if you negotiate a bit better, they are incentivized to tell you not to negotiate.
No. That's not how it works although your analogy is sound.
When a company hires a recruiter their job is to find a person for a fee that may be fixed or not. In that case the recruiter works for the company much like a real estate agent works for the seller.
Other recruiters work more like a buyer's agent. The job seeker hire them and they find positions. That kind is much more likely to give good advice about negotiating power.