Wondering how this turned out as I'm having the identical problem and I have a hypothesis if you'll indulge me.
I'm having this same issue with a Briggs power washer. I'm starting to think it's a faulty auto choke. I'm trying to find out how long it should take for the auto choke thermostat to fully actuate but can't seem to find the information. I've timed the one I have and it's anywhere from 3 to 4 minutes, which seems excessive.
It appears as if the choke spring holds the choke closed, not open. So a tighter spring may not solve the problem. The O.P. said that this happens when the blade is engaged. I have the same symptoms when I engage the water pump.
There's no throttle on this so it runs full. When not under load, the governor is putting closing pressure on the throttle, which in turn, puts pressure to open the choke (the way it's designed with the oblong interference linkage). When it comes under load, the governor releases the pressure and the governor spring opens the throttle, which takes the pressure off the choke lever allowing the choke spring to close the choke unless the thermostat is putting pressure back on the choke arm.
This appears to start the oscillation as the airflow is interrupted and the governor tries to stabilize things. Because of the way it's designed, this starts a sort of oscillation causing the choke to flap, until the mechanism of the thermostat kicks in and puts permanent pressure, holding the choke open. If I take a screwdriver, hold the choke open, and then engage the water pump, everything runs perfectly. This appears to be why after it warms up (almost 5 minutes), everything is fine. But I think the problem is that the thermostat choke is faulty in some way and taking too long to fully heat up and engage. Does this make sense?