I dont want the thread to be completed without that information. Yes it is possible that a transformer can get more heat in no load and low heat in load condition. It was nearly like a CT, but a potential transformer.
Assumption:
A step down transformer with comparatively very high primary winding resistance, very few core segments but not single, low resistance secondary.
Operation:
A very high primary resistance in the transformer can be considered as a resistive load instead of inductive load. so the current through the primary winding is mostly depend on the winding resistance instead of flux in the core.
So this transformer can be assumed as a multi turn primary CT transformer with highly resistive load in series. hence the transformer is designed to provide the flux enough to produce a full load current in rated voltage, it will produce more flux in the no load condition and the flux will reduce in loaded condition due to secondary flux.
A high value of flux may produce square of eddy current loss in the core but the load will reduce the flux and also the eddy current loss. but the loss in secondary winding is low because of low winding resistance. So now total variable loss is increased and the constant loss is very much decreased.
(obviously In this case we cant tell the eddy current loss as constant, because the cumulative flux in the core is not constant)
due to more loss in no load condition, the transformer is heating more. but this is a poorly designed transformer because high power loss in primary winding, the current taken from the primary is not relevant to secondary, so it will take unwanted current when there is no load in the secondary.
Advantages:
Small size, small power so small loss in power, less complication in design and low cost.
But I have totally made assumptions based on my theoretical knowledge, So im not that much particular about the amount of change in primary current.
As for OPs concern, no need for confusion and just select a good transformer of some more power rating.