We've stayed at the DLH twice in the last few years, my son has to take various refridgerated meds inc daily copper injections, they have a fridge at the luggage desk and all we do is put his meds in a small cool bag and they just put a tag on it with our details, each morning and evening we get his meds, do what needs to be done and take them back to the desk where they stick it back in a proper fridge.
We did consider the minibar thing but its not cold at all, I mean really not cold, more like room temp on both times we stayed, as our meds need to be below 5/6c a proper fridge is the only answer.
On the travel to Paris on the Eurostar, it's worth asking your local hospital or even the GP practise for some cool bags, once frozen they stay cool for most of the day and we just stick one in the cool bag with the meds and its always still very cold some 7/8 hours later.
Hope this helps.