Silicon chips are normally 125C junction. If it is thermally protecting itself it is shutting down and not running.
Normally all you do is add the thermal resistances, multiply by the wattage, and add to the ambient. (22+11+1(guessing, it depends on how good the interface from the pad to the sink is, and I am assuming you have a sink pad since it gives a junction to case))*1W+85C=119C dangerously close to 125C but doable.
For a peltier you are looking at your power plus the peltier power going into the heat sink. you are probably talking about 4W going into the heat sink for 44+85=129C at the peltier. if you get one that will get you 40C down you are at 90C. either way you need a better heat sink or a big peltier.
Dan