For years nobody has came up with a reasonable explanation as to why the 2015 only has the KEY and DISP tests but lacks the well documented* K2000 BUILT-IN mode. Keithley and Tek have not been forthcoming with the info when asked either. Everyone outside of Keithley has resigned themselves to no proper tests

Some had suggested there was no room in the ROM for the built-in tests due to the THD board. But anyone looking at a ROM dump can see there is plenty of room. A 'strings' dump shows the tests are still in there. But that could just be left-over garbage from K2000 code. What convinced me is that there are 2x TEST menus in the ROM. One with just the TEST: KEY and TEST: DISP and another one with the same strings again but with TEST: BUILT-IN too. Why have 2x menus? What a waste of bytes?
So I just spent the evening playing with IDA and to cut a long story short it has been staring us all in the face for some time, it is as simple as...
K2015 powered off. Hold the SOURCE button and power on. (
The same procedure as documented in the calibration manual for doing a factory cal. 
)
Now press SHIFT and TEST and there is
BUILT-IN flashing in all it's glory.

Sadly my K2015 fails on the same 306.2 and 306.4 errors that I have on my K2000. I know this is not considered a fault by Keithley as long as the meter still meets specs and is probably why they hidden the damn' test after they introduced the 306 bank anyway

Time to replace some JFETs...
FWIW from my IDA investigation I found that the keyboard/display to mainboard interface appears to be using nothing more fancy than simple serial TTL 9600,8,N,1
(* Well documented, except for the bloody 306 tests

)
ETA: Firmware B17 here.