I've never found anywhere outside the US that takes my Discover card.

Yes, Discover is accepted, but only at a few places. If the terminal is BDO, it should take Discover.

I wouldn't risk it. MC and Visa are your best bets. Amex isn't a sure thing either. Only use ATMs INSIDE banks or hotels.

Never use or even have a debit card….

Since you mention Discover and TD Bank, I'm guessing you're in the US. Open a Schwab brokerage account along with their free checking. Use the checking account's ATM card for free withdrawals at ATMs in the Philippines. Also, if you can, get a Capital One card to use at most businesses — it protects your debit card from fraud and avoids foreign transaction fees.