A shorter scale neck won't cure the floppy and buzzy string problem. From a string gauge standpoint there's no difference between a shorter neck and a regular-length neck with a capo. Either way you'll need to use heavier gauge strings to achieve your objective. Of course, if you do find an instrument with a significantly shorter scale-length, the builder will have already worked out what string gauges to use so you won't have to experiment.

And I'm afraid you'll find that most non-guitar-neck-on-a-guitar-body instruments are, indeed, custom made. The only off-the shelf one I'm familiar with is Gold Tone's "Dojo" which is a 5-string banjo neck on a resonator guitar body.