I don't know of any way to adjust the rain gauge in WC, but I do question the "
lubrication" of the tipping bucket. I can not swear to it, but I don't think the Davis 'bucket' had any "lubrication" on it when I bought it some 5+ years ago. I am positive that I have never read any instructions that it should have any "lubrication" nor have I done anything more than clean out some sediment in the buckets.
Many kinds of "lubrication" are petroleum based but even other types can be affected by temperature. Usually that means a thickening of the lubricant. In a tipping bucket (seesaw) design, where extremely small weights are concerned, that could easily cause the bucket(s) to delay their normal movement resulting in a lower than normal rate of movement (a few extra drops required for each 'tip'). While that could cause a lower
amount report (more drops remaining in the bucket when the rain stops), it would also result in lower
rate reports.
On the other hand, perhaps your Tesa hardware is different, but I would suggest checking any manual or contacting their support documents.
On the third hand, I wonder if many of us maintain two different methods of measuring precipitation (or more than two identical hardware kits) so that we might be aware of any differences. Those of you that use the standardized, manual, calibrated rain gauges certainly have an advantage there.