The problem with the oil is that it will get into the mesh that wraps the wire in the probe and then will the heat shrink tubing stick to the oily wire?
These probes have thermocouples on the end that measure the change in resistance (I think) as the temperature changes. The probes all look alike in my unscientific research, but the boxes with the meters are sometimes the same, sometimes different and independent of whatever label they may have. I would guess that if your meter box looks the same as the replacement, it might work OK.
Some of these say they are guaranteed for life, which means if you pay the $15 to cover shipping and handling, and you spend $5 to ship it back, they will send you a new one that cost them total maybe $5. Just like ink jet printers and the ink.
What this world needs is a good external reading thermometer. That will not die just as you need it, and you can buy reasonable priced replacement probes, maybe even by the dozen.
Maybe CS could do the research and private label their own. Something is the range of $20 to $40 seems appropriate considering how important they are to good Q, and like $5 each for the probes unless you buy them by the dozen.