Background
Suppose I look into a EUR-vs-JPY Cross Currency Basis Swap with MTM feature (i.e., including quarterly resetting notional on one of the two legs). Cashflows are projected off the ESTR & TONAR curve, respectively (with the basis being added on one of the legs).
Now, I assume that my trade is cash collaterized exchanging EUR-ESTR. Then it becomes clear that for discounting my cashflows, I will use the ESTR OIS curve for the EUR leg and a CSA-adjusted curve for the JPY leg (accounting for Euro-Yen basis, crossed via USD), just like e.g. BBG SWPM suggests.
Question
Which curve should I use to estimate my discount factors for the FX reset projection when valuing this deal? Intuitively, I was thinking to use "all I have", i.e. fx swap curves that I fit so they match USDJPY and EURUSD pips that I observe in the FX forward market... However, it seems to me that the market does not trade these products based on any "default CSA assumption". That said, using such a curve, I would be missing out on the EURJPY basis, or no? Would Fx resets instead be estimated using the very same CSA-adjusted JPY curve I described above for discounting the JPY cashflows? What is the market standard practice when building such a curve -- does one mix Fx swaps in the short end and Xccy basis swaps in the long end, although they have different inherent CSA characteristics?
