It's all down to temp differences. You've got incoming water temp of say 4 degrees and an internal temp of 18 degrees. That's 17 degrees difference. Your analogy of the cold drink on a hot day is a good one. You can't beat the physics. You can twist them though.
I assume the new cistern is a dual flush low capacity item so don't line it with polystyrene insulation because it won't have enough water to flush properly. It's also probably thinner ceramic than the old one so it affected quicker by temp changes.
Set the radiator to 1 or 2 on the TRV. To be fair there's no reason to have a rad in there seeing as it's such a small room. It will be cooler than the rest of the house but then it will also reduce the temp difference.
Close the door.
Don't have the handtowel(s) drying on the rad.
Now a final option would be to swap out the extractor fan for a heat recovery one. A wall mounted one will be £300 odd though but it will keep the water vapour levels in check.