One way to think about this is as an acquisition rather than a merger. In that sense, your company would be acquiring the assets (or stock) of the other company. You would make an offer to purchase this individual's company, perhaps in exchange for stock in your company.
The result would make you partners with this individual and it would functionally merge the two companies into one. If you were to think of this as a pure acquisition, you could break down what assets or items of value you would be acquiring from this person. Customer lists, contact lists, contracts, an agreement for the founder to work for your company for a given amount of time, perhaps attached to a vesting schedule for the stock purchase pay-out.
In this sense, I think it changes the conversation a bit and allows you to actually construct a tangible offer for the other founder to consider. Perhaps you would make this person your VP or put them in another executive position after the merger goes through, but they would clearly know what they are getting for what they are giving up. In the end, the resulting company should have a singular mission and everyone would be aligned underneath it. I think this type of offer clarifies the opportunity for all involved.
I think the issue of whether this person want to throw in with you or keep trying to make a go of it alone will still be the same, but a formal offer for acquisition puts some teeth to the concept of a merger and starts talking about exchanging value for value on the table more clearly.