In fact the pilot in question (with a multi-engine ATPL) needs not attend any formal training school such as FSI.
By FARs all the pilot needs to show is knowledge of the aircraft systems and three takeoffs and landings to a full stop.
Now that is by FARs, in the real world the insurance companions really control who may fly in the cockpit as PIC or SIC. Most insurance companies, possibly all, do not require a type rating for the SIC, however, formal training at a school such as Flight Safety is mandatory with most (the great majority) of insurance company underwriters.
One point, you stated that some flying may involve European operations. If so, you need to check the SIC requirements of each country you will be operating in, a point is case is the UK which requires both pilots to be type rated.
This is for Part 91 and 135, however, the FAA does require additional training for Part 135 operations. However, this additional training for Part 135 would be address at a formal training school such as Flight Safety International.