You can make a Kake Kutter by sawing the base off of a fired .30-06 shell and chamfering the case mouth.
If you really have to have one, the 310 Tool boolit sizing chambers come up pretty often on the auction sites. The most common are 0.311" and 0.358" (nominal). These were designed for plain base boolits, or, at most, gas-check designs with the slip-on Ideal gas checks. I don't know how they would work with the newer Hornady crimp-on type. They are pretty much for the subsistence reloader, who loads a box of shells a year; kind of tedious for the typical modern reloader who wants a reasonable number of reloads for the time expended, but the "experience" is always interesting, of course.
You might also look for a replacement 0.311" expander plug for your expander die; it should have a belling flare at the top which might make seating the cast boolits easier at the next step. The 0.308" expander, as you say, is for jacketed bullets.
The dies don't have to screw all the way to the bottom of the spigot on the tong tool. Screw the muzzle resizer in until the neck is reduced to the depth of the seated boolit and lock it there. Lock the expander die just at the point the case mouth is slightly flared, and make up a dummy cartridge and set the seater die and seating screw accordingly and lock them in.
You should have no problem using the tool if you can find dies for the .270 Win, the .280 Rem and other shells similar in length to the .30-06. Shorter, like the ,250 Savage or .243 Win, might or might not work, depending on how long the spigot on the tongs is and how deep you can screw the dies in. It's just a matter of trial and error.