Being a 1A starting serial number, what year is the boat?
Being a newer engine model, something's have changed as far as sensor designs and unfortunately there isn't any updated wiring diagrams available online along with a lot of the testing procedures found in older manuals are almost sort of invalid such as in a situation like this. The older IAC's were a 4 pin while the newer ones are 2 pins. The old TPS's were 2 or 3 pins and now has 4. Wiring diagrams for the MPI system is no longer found in service manuals any longer, they are in paper form which is sent out to dealers as they are produced for a certain engine model...This stated in 2007.
If your lowest RPM you are truly able to obtain is 650, then that is still to high. Before 2006/07, the MPI idle speed was 600 but the later models after 06 or 07 went to as low as 540 RPM when engine is warm at 160 degrees.
Were you ever able to get the engine to die on the garden hose at home or only while under load in the water? Since it seems to be dying during shifting, it could have something to due with the mercruiser shift interrupter switch binding up which affects the ignition system or more probable something different is happening that is just out of range to make the engine not run correctly but not bad enough to set a fault code sounding a warning horn. It could be anything from a sensor to a throttle cable not adjusted correctly holding open/closed too much with the throttle body plates....Also possibly affecting shifting....Among other things as well.
The MPI system may need to be scanned by someone that knows the system pretty good to see if any of the live data parameters seem out of specifications.
You could try pulling out a few easy to get to spark plugs to see if they appear to be overly carbon fouled since you mentioned the MAPT sensor which yes if there is something not right with the MAP or engine vacuum systems, then it could have an effect with what you're describing.
Being that you have the flat distributor cap, it may be causing issues as well or the distributor shaft assembly may not be properly phased in too which will cause issues too.
Is the boat new to you or did this just seem to start out of the blue?
Davidalex's IAC pin connector would resemble this as it matches his part number,
http://www.ebay.ca/itm/Sierra-Idle-Speed-Control-Motor-Replaces-Mercruiser-862998-/322204451365