If a US entity borrows in EUR and I need to perform a DCF valuation on that borrowing, should I use USD based curve (for the appropriate rating) or EUR based curves? In other words do I use the domicile (US) or the currency (EU) of the entity/loan to determine fair value? I believe in derivatives we discount based on currency and the use spot rate to convert to USD.
You should use whatever currency in which the debt is denominated. Specifically, since it is the EUR currency and interest rate risk associated with the debt, some sort of EUR curve should be used.
Theoretically, if you are looking for the present value in USD, although the debt is denominated in EUR, you could convert future payments at the forward currency exchange rates and then discount at the USD curve, but it would be more simple to just discount using the EUR curve and then convert at the spot exchange rate.