You can gain about another 30deg of view by buying a wider lens. Something like a Sigma 10-20 or Tokina 11-16. Priced in the $500 to $600 range. But they will not give you the 3 shot wide effect you are looking for. Fuji and a couple other companies did make such a film camera. But it was done with a near fisheye lens that image circle was way larger than needed. And the film only recorded the center width of the image circle. You can do this with a fisheye lens and cropping in an edditing program.
It is much easier to use a wide lens and stitch though. As you will get distortions with the fisheye's (at least with the lower cost ones). There was a post on here not too long ago about a custom made nikon fisheye that had very good control over unwanted distortions 13mm mf lens. But its over $6,000 used. That would be a prime candidate for what you need. over 180 deg of coverage with 118 deg of diagonal coverage. It requires careful use though. As tilting or out of level increases distortions.
So the lens does exist! Not sure how well it covers on a crop sensor though.