Some thoughts. The fault sounds like its in the input protection. Idle for a few days improves performance suggests its the PTC. The photo looks like a PTC, but I don't know the 787 PCB, but the input circuits of Fluke HH appear similar.
Try forcefully cooling the 787 with freeze spray to R63 or putting the whole unit in the freezer. If it improves, it again points to something thermally resettable. If its truly the PTC, Fluke PTC parts are typically 1 k 2 W but I don't know the exact part. If you substitute it with any 1k 2W resistor and the meter functions correctly, you have the right part to replace. Older Fluke's used a fusible resistor instead of a PTC, and you can use that until you find the right part.
Note if the PTC isn't the only thing faulted, look then also at the MOVs, as they tend to get damaged together by some issue.
Looking at the second image, the markins are DALE R301. If you look at this document, the code is for a 301mOHM resistor, which matches your measurement.
I attached a picture of the 787 from a while back.
As saturation mentioned, the PTC should measure around 1k ohm. The black colored PTC is labeled RT2 (right hand side of the picture). Just below it is a green 1k ohm fusible resistor. The blue MOVs are labeled RV1, RV2 and RV3. These MOVs should measure 0L (open circuit) ohms. All these measurements can be done on the 787 "in circuit".
The power resistor next to the green fusible resistor is likely a current limiting resistor.
PS. Are resistance measurements on the 787 accurate? For example, does a 1k ohm resistor measure 1k? How about a 10k ohm?
Thanks for helping me,
I did some measurement and here is the result.
The black PTC : 1.146k ohm
Green fusible resistor: 1.011k ohm
Brown color current limiting resistor: 0.910M ohm
Blue Mosfets: All of them are fine, OL ohm
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Something really odd happened when I used the 787 to measuring resistance
I used Fluke 702 sourcing 2ohm to Fluke787 and the reading is 0.3ohm
Sourcing 0 ohm to Fluke787 and the reading is -1.6ohm
Sourcing 10k ohm to Fluke787 and the reading is 9.76k ohm
Sourcing 1k ohm to Fluke787 and the reading is 1.004k ohm
Finally, I used fluke 1857 to sourcing 50V DC to 787. I don't know why but the sourcing voltage drop to 17V and 787 showing a reading of 12.3V
I used a fluke 179 to verified the fluke 1857 is working fine.
I used Flir E4 (hacked to E8) to inspect the pcb at the time fluke 1857 sourcing 50V dc to 787. Nothing getting warm!
Please advice,
Thanks