NEVER run a handheld multimeter from a bench power supply or wall wart. Why are people so cheap.
Because input com(-) is not DC power(-) and any short or voltage forced between them will blow up the main multimeter IC or diode CR5. Check CR5.
For the Fluke 87III, I'm not sure if I have the specific service manual for it. edit: 87IV,V I think: It has three voltage regulators (power rails) U10 making 5V, U6 making 3.3V and U8 making 2.5V. BUT to get +/-2.5VDC and +0.8VDC, they use a floating common to the +2.5V regulator as analog common. Yes it's confusing.
Running the meter from a 9V battery, check the three rails with another multimeter its (-) to the 9V(-).
Should measure around 9V at TP16 U10 input, 5V at TP17, 3.3V at U6 output or C13, 2.5V at UI8 output or C16.
added block diagram and typo up top? VDD should be +3.0V not -3.0V I think.
I would think the main multimeter IC is blown, it usually gets damaged because it's only a 5V part. Or dual diode CR5, or Q8 have failed but the power would not switch off.
Power Switch
Q3 NPN BJT, MMBT3904, Fluke 742676
Q8 N-ch MOSFET, 2N7002, Fluke 927538
Q10 P-ch JFET, SST175/J175, Fluke 832477. SOT-23 DGS, SMD marking code "S5" or "P05".
CR5 dual diode, BAV99, Fluke 742320
Input Protection clamp
Q13,Q14 NPN BJT, MMBT6429, Fluke 912469, SMD marking "1KM" or "M1L".
Q1,Q2,Q6 NPN BJT, MPS6560 but also given as SMBT1388LT1, hand selected for low IEBO, Fluke 821637
Backlight etc.
Q5,Q11 PNP BJT, MMBT3906, Fluke 742684