A couple of observations for you:
1) You can simply remove the super cap and clean the board area around it. The meter will be nearly fully functional (only some memory/RTC functions use it) and you will be able to do all of your tests and work without replacing it.
2) Your erroneous resistance measurements are high, not low. Input protection MOVs and other components cannot be the cause of high readings, only low.
3) The accuracy of meters is not that good at high resistances, from 5M to 50M it is 1.5%, from 50M to 100M it is 3% and above 100M it is 8%.
The first things to check when you have an issue like this are the test current and compliance voltage. So take another DMM that has a 10M input impedance, connect the two together and set the 287 to resistance and the other to volts and then to current. See if those output voltages match up to what is shown in the manual. High resistance readings only in the upper parts of the range is a pretty strange symptom, IDK offhand what would cause that.