If the BZ07 has a cartridge heated group, and the problem of Bas is kind of the opposite, the group overheat to kickly, how about a thermostat with lower temperature at the group? Maybe around 75C, so the group cycles between 70 and 80. If you look for "KSD301" at ebay.com, there is a lot of this small thermostat from Hong Kong...
Well, this could help with the flushes, but maybe not with the rising temperature during shot. For that, maybe only this great mod Eric did on his BZ35 could help.
Or, another idea, but much more drastic: a cold water input near the water inlet of the group, maybe with a solenoid or maybe with a check valve, and a needle valve to regulate the cold water flow. Something like the thermosyphon when pulling a shot but without the fuss about how the loop heats the group, just the new variable "cold water flow"