Single Cut 4 string kit

I'm afraid I've never seen a single cut bass kit either. Your best bet might be to buy an unrouted LP or Tele body and fit a neck to it.

 

You might consider buying a kit from Aliexpress. Here 's a 4 string single cut, not a kit, but most of the sellers can get you almost anything you want. The basic woodworking on the Chinese basses is ok, its all CNC. Where they cheap out is hardware and electronics. The frets usually need a full level and polish, but that would be true of almost any kit. Another option would be to buy a "paddle" style unfinished neck from Carvin/Keisel. The fretwork is done, but the headstock is just a rectangle ready to be custom shaped. Building a body isnt hard, but you need access to some basic woodworking tools. If you just want a traditional neck, plenty available through Allparts, Mightymite, or best of all Warmouth.

Online Shop Custom Shop 5 String China Electric Bass Guitar 24 Frets Ebony Fingerboard Guitar Neck Left Handed Guitarra Body & Kit Avaialble | Aliexpress Mobile

 

Another option:
Warmoth will sell you a body blank already routed for neck pocket. You could have them route for pups, electronics, whatever, then all you have to do is cut out body outline and bolt on the neck of your choice.
Warmoth Custom Guitar Parts - Custom Bass Body Wizard

 

What kinds of tools and skills do you have available? Pretty much all of the support you'd need is right here in this forum. A neck can be had with just a simple maple glue up, rout for a truss-rod, using an existing pre-slotted and radiused fretboard from LMII. Using an easy Fender style headstock can simplify the complexity of a neck. We can also support you through designing and building the body and mating them. You can do it!

 

I was linking to the square body blanks, pre routed with a neck pocket. Dont know if they have enough meat where the single cut upper horn would be, might be worth a call to Warmoth to see if they can extend the blank. The Warmoth necks are not cheap, but very high quality. Good luck with it!