Coleman 400 cut out when hot under load ideas?

[Homeowner A]

Have a Coleman UTV 400 that cranks, runs, drives well until it gets warm and is put under load.  Light throttle or full throttle it doesn't bog, but it will cut out mid throttle if you move the pedal quicker than it seems to like for example when you're getting ready to go up a hill, but if you immediately floor it, it will do just fine and not cut out.  It's random if and when it's going to do it, but is consistent in that it happens only when hot/full operating temperature.  I've replaced the injector twice, same with the spark plug and tried different heat ranges (NGK 7,8 and 8 iridium), replaced the fuel pump, adjusted the valves, checked air filter which is like new, changed the oil, replaced the fuel filter.  No check engine light.  I don't have a code reader for this.  Initially I thought it might have a head or head gasket issue as when I got it, the oil was over full, very runny like water milky and it was smoking.  I've changed the oil since then twice to clear and clean everything out, and it seems to be good.  Coolant stays the same level and so does oil with no changes in consistency or need for adding more. Had I think 7-9 miles when I got it and now has about 77.  What should I try or consider next?  

[Ben1098]

Past experience for a random (as in your heat soak cycle-----engine hot and will come back to life again with some cooling) is the pick up coil.  Next is the ignition coil.

Use a spark gap tester and watch the spark quality and/or hit and miss or dead.  The really OLD timing lights (60's) used just a dim neon tube.  A weak spark would result in a really dim or non flash.  Also, Snap-On sold a handheld spark voltage tester with flashing LED indicators and a thumb wheel voltage trip point selector.  Lab scopes give a visual check if you know a friend who has one with the spark pick up clamp.

STORY:  I have had some pick up coils that would take forever to fail and then heal BEFORE you could pull the connector and connect an ohm meter to check for an open coil instead of the spec'd resistance.  That one was solved by using a DPDT switch to check the resistance right as it died......could watch it "heal" again in less than 5 seconds.  It had been thru a bunch of shops before me.