when using RTOW charts how does one account for a forecast temperature inversion?
One needs to look at both Hp and OAT at the inversion. Depending on the presentation of the chart and the inversion height delta, you might either correct both for pressure and temperature or run the inversion temperature back down to aerodrome level at a suitable lapse rate.
Main thing is to be aware that a significant inversion is going to present you with problems if you lose one during the takeoff so you should do something sensible to keep some extra (T-D)/W up your sleeve for the possibility. If your company procedures don't address the problem, then the above suggestions will give you a fighting chance.
Derated-TO is a max. thrust.
Doesn't matter which brand we are talking about. Flex is reduced throttle, derate is a defacto lower thrust engine bolted onto the aircraft. Flex plus derate is a combination of the two.
With flex you can run the thrust back up to the (relevant) derate or max thrust setting but, I suggest, slowly to avoid any nasty surprises.
With derate, you need to be aware of the very real potential for Vmcg/Vmca problems if you increase thrust significantly during a low speed schedule takeoff. This latter is quite a probable situation for those operators which use derate to get a better RTOW out of shorter contaminated runways - the reduced Vmcg/Vmca permits a lower speed schedule to fit better with the runway length.