Well, the low duty cycle sure explains why those average voltages you measured were so low
Looks to me like the computer periodically applies current to the sender/trim pot during ADC sampling. Could be through a transistor or other type of analog switch. May be part of a multiplexing arrangement due to single ADC.
One approach would be RC filter of PWM and buffer with an op amp. I’d avoid having the op-amp connected to sender wire directly. Using a series 71 ohm resistor, the op amp would need to output 0V for empty and approximately 2V for full.
Op-amp needs to be able to sink 14mA and could be powered from the 5V and GND available at TB1 positions 3 and 4 of the LCD brd. VR2 and PWM max still used to tune the empty/full range.
The TLV2472 used on the LCD brd would work. A single version is the TLV2471
-Chris