Thank you for the PuTTY log. It showed that PuTTY uses two step public key send routine, while Rebex is using one step routine. I have tried to add the two step routine.
Since I am not able to find your active support contract, I am sharing here a link to TRIAL version of updated code: RebexSshPack-BetaBuild6150-Trial-Binaries.zip
Please let me know, whether this BETA build solved your issue.
At version 2017 R1 we added
EnsureKeyAcceptable option that instructs SSH client to announce public key to the server before performing key authentication (two step routine mentioned above, default is one step routine).