That's not too bad.
One thing I experimented with was splitting the Mild (18-10 °C) category in two in order to fix the Cc distribution. I found that making the "sub-Mild" category 10-14 °C still made Cc too extensive, but 10-13 °C seems to fairly closely match the distribution of Cc climates. Including that could be a viable alternative to making a separate map to sort out the b/c boundary. That way you'd only need the temp maps to get the Cc distribution right. It would however ramp up the temp possibilites from 100 to 121.
Here's the July map with the 10-13 °C sub-Mild category (in light blue). It also has -3 to 0 °C "Chill" in grey but that's not a necessary addition. If you compare the extent of the sub-Mild category to Cc/Dc distribution in British Columbia and the British Isles, I think it's a fairly close approximation. Seems to work pretty well for Norway, too. Though some areas become Dc instead if using the 0 °C isotherm (those maps use -3 °C).
Here's the map:
JulTemp Test.png
Also, I think maybe the Db/Dc distribution could potentially be improved by changing the Cold/V. Cold boundary from -10 °C. At least in Europe Dc is somewhat less extensive than what it should be in the present model. I haven't experimented with that yet, though.
Edit:
I experimented a bit with -8 °C and -6 °C Cold/VCold boundary. Tbh the results are much less promising here than with the Cc distribution. The Dc distribution in Scandinavia would be more-or-less correct with -6 °C, but then you end up with northwest Russia and Eastern Canada having too extensive Dc areas. I ended up with -8 °C as a compromise. The results might be slightly better than the original -10 °C boundary, but I'm not sure. With -8 °C the Dc areas in mountainous regions become even more extensive than previously, although in reality the Hindu Kush and other similar cases are mostly Db.
Climate zones with sub-Mild (13 to 10 °C) and -8 °C Cold/VCold boundary:
13to10 Sub-Mild and -8C Cold VCold boundary.png
In the end it might still be easier to create a separate map about the length of the growing season, since it doesn't look like the Db/Dc boundary can really be sorted with info from just two months.