Is there a minimum weight requirement for flying manned aircraft in controlled airspace (B, C, D, E)?
If there were any minimum weight requirement for manned aircraft in controlled airspace, the requirement would probably be in 14 CFR Part 91, Section 126 through Section 135. These are the sections which describe the operating rules for various types of airspace. I looked through those sections, and there are no weight requirements.

I also can't imagine why there would be a minimum weight requirement for any class of airspace.

Like most aircraft, the Colomban Cri-Cri must have an airworthiness certificate and a registration certificate (14 CFR 91.203). The regulations describing getting an airworthiness certificate are in 14 CFR 21, Subpart H.

Some aircraft weighing less than about 254 pounds are "Part 103 ultralights" and do not need to be registered. However, this doesn't apply to the Colomban Cri-Cri, because Wikipedia says that it has a cruise speed of 100 knots; Part 103 ultralights must be "not capable of more than 55 knots calibrated airspeed at full power in level flight".