More than that it is more likely to deter than encourage use of the coop. Your coop is small and has fairly limited ventilation, their bodies alone will generate more than enough heat - adding additional heat into such a space is likely to make the coop more uncomfortable to them.This is unnecessary. The chickens will be fine without the extra heat. If anything, the added element of the radiant heater could be more of an unnecessary risk.
I think raising the roost will help, but they definitely like it up high in the trees if they can get in them. If you can find an old shed, or build one, make sure it's high enough with some roost bars to make them feel safer and I think that will get them out of the tree where they are more susceptible to other predators!!