If you can get the BIN tinted to the colour you want, just use two coats of BIN. There is no need for anything else. BIN is alcohol not oil based.
I have used oil based eggshell in problem areas in the past but it stinks for ages and eventually yellows.
With regard to protection from water and moisture, oil based paints win, followed by BIN and then followed by waterbased eggshells.
Waterbased eggshells are normally sufficient in bathrooms but it sounds like your bathroom lacks adequate ventilation. I am suggesting a belt and braces approach, you could however treat the crazed areas with either (off the shelf) BIN or oil based undercoat (as recommended by your decorator friend) and then just use your existing emulsion over the top of that and see how you get on. Now that the weather is cooling, you should see if the latter option is suitable within the next 2 or three months.
The cheapest option in the short term is the oil based undercoat and emulsion, it may or may not, however turn out to be the cheapest in the long term.
Good luck either way.