I'm sorry to say this, but this employer sounds like a scumbag. If this is how he treats you (by lying and trying to exploit you) at this stage, how is he going to treat you once you're stuck with him?

> he has to pay thousands of dollars to hire a lawyer to include the terms

This is utter bullshit. Call 5-10 local lawyers, tell them what is needed, and find out how much it would cost. Present the employer with the names of those lawyers and their prices.

> Is my request unreasonable?

ABSOLUTELY NOT!!! Always get everything in writing, no matter how much you trust someone. For all you know, the person who made you the promises will be at a different company or dead in a few years.

It's completely normal and reasonable to get something in writing, and it's either naive or malicious that someone is refusing.