Generally speaking you lose every time you convert currency. If you can get the JPY into your card as JPY (it seems to depend on where you buy the card whether that's possible or not), then you're probably be best off using that.
Travelex say they take a 5% or 5.5% (depending on the website) cut every time you do a currency conversion (that's their cut, which is on top of the interbank rate). Since you want to use it in Taiwan, and I don't think they support TWD at all, then you're stuck losing 5% or 5.5% if you want to use the card. If you lose another 2.5-5.5% converting the JPY to USD you're going to be down as much as 10-11%.
ATM cards, at least at my bank, take about a $4.5 USD fee on top of the ATM fee plus 2.5% for currency conversions, so I would not benefit from such a card (break even is at a withdrawal of less than $200).
If you happen to be going through HK on your way to Taiwan, there are good options for cash currency conversions.
The above does not consider currency exchange rate changes during your trip- generally the USD has been strengthening lately, but that's not guaranteed to continue. Charts are available from xe.com.