Sounds similar to a self-funded search where you find a business and then raise capital. Certainly not a traditional search fund, but definitely part of the broader category of entrepreneurship through acquisition. You will find that some of the traditional search fund investors are open to investing in these types of deals.

I would just say if you have a target identified there is no need to offer the "Step-Up" to investors ala the typical search model. You already did the hard work. Having a deal signed up is very strong currency in today's market and there are many groups (like mine) that spend vast amount of marketing dollars and times trying to find these exact situations.. we typical refer to this area (non formal search funds) as independent sponsor if you search that term.