Unfortunately, the distilled water is likely to produce a very flat-tasting tea.
Are you sure
they won't let you install a faucet filter on the bathroom sink? They are very easily installed, and can be installed & removed without damaging the sink (as long as you have kept all the parts). It only took me about 5 minutes to install our first one (and about 10 minutes to remove it when it turned out to be broken and needed to be sent back).
Normally, I would recommend a PUR 3-stage filter, because it adds minerals back to the water (and is more effective at removing contaminants than most other small filters on the market). However, they don't offer their 3-stage filter in a pitcher format.
I guess your best bet at this point is to buy a Brita or Pur (doesn't really matter which) water filter pitcher and keep it in the fridge. As for which water source to use (the water fountain or the bathroom sink), I would recommend that you try running both through the filter pitcher, and stick to whichever one tastes better after filtration.
I would also suggest that you consider moving out of those dorms at your earliest opportunity. No kitchen sink? What a joke!