Should there be any problem cooling Mysore Pak outside a refrigerator? I am interested to make Mysore Pak, the Indian sub-continent lightly-roasted Besan (Chickpea flour) based cake.

I can indeed lightly-roast one cup of Besan, add one cup of sugar directly together with one cup of water, stir all three ingredients to a fine smooth batter and and then add about one cup of coconut and/or sunflower oil and mix well a few minutes to get an extremely unified batter (with no "lumps" or "chunks").

My problem is with cooling off the batter until solidification (which is how the batter is made into a cake):
The refrigerator in the apartment I rent is a bit small for the pan in the sense that even if it is well situated it will probably touch plastic surface, walls and dishes and that I want to avoid.

Should there be any problem cooling Mysore Pak outside a refrigerator?
Batters are not shelf stable. It doesn't matter if their ingredients are shelf stable by themselves or not. So you have to follow the basic food safety rules for non-shelf-stable food. In particular, you have to ensure (no matter by what means) that it doesn't spend more than 4 hours with an internal temperature between 4 and 60 degrees Celsius. 

If you can prove that your cooling procedure does that, you are fine. If you don't know, then it is unsafe.