No.
As much weight as they carry, so far anyway, an insurance requirement does not constitute a "regulation" for the purposes of the 61.51(f) SIC logging requirement that
"more than one pilot is required under the type certification of the aircraft or the regulations under which the flight is being conducted"
IOW, an insurance requirement is not an FAR or a mandatory OpSpec requirement.
There may be some other basis for logging SIC time, such as an IFR passenger flight under Part 135, but for plan vanilla Part 61 ops, no.