I'm afraid that's just a sustain pedal with TS (tip, sleeve) cable connection. A halfdamper pedal has a TRS (tip, ring, sleeve) connection and sends continuous messages from 0-127 like an expression pedal.
As there doesn't seem to be too many half damper pedals by different brands, a possible solution could be an adapter which inverts the polarity. I don't know if anyone produces such a product, but if it exists it could be used with the headphone output on the Rev2 as well, to switch the left and right channels so that the headphone output reflects the other outputs, and not a mirror image as it is now.
Coupling two Y-adapters should also do the trick, one that breaks out the tip and ring to 2 mono cables, and then another Y-adapter which merges the 2 mono cables back to a TRS connection. Just standard stereo to 2 mono Y-adapters. Connecting these adapters together left to right, and right to left, should work.
This one connected to the damper pedal:
https://www.thomann.de/se/pro_snake_tpy_2003_bpp.htmThis one connected to the Rev2:
https://www.thomann.de/se/pro_snake_tpy_2003_jbb.htmAs they are marked with and black, connecting them together black to red and red to black, should invert the polarity the way you want.
However, I have never used an electronic half damper pedal, so I don't know exactly how it works. But from what I've read so far, as I understand it, I don't think it's possible to get the same results when using it directly with the Rev2, as when using it via your Yamaha. In that it can't be used for sustain, but only the continuous expression message. And I don't think there's a workaround for that, other than going through the Yamaha. As the Rev2 isn't designed to work with a half damper.