Visiting the Philippines as a dual citizen with a different name in passport?

You have a Philippine passport, that means Philippine immigration will let you in. They're not really going to care how you got there and they'll not look at the tickets. Philippines customs might care where you came from but they're unlikely to look at the tickets either.

The problem is persuading the airline to let you on the flight. They get charged if you're refused entry so they'll want to see evidence you'll be allowed in. However since your R.O.C. passport has both names I'm sure you can show them both passports and explain the situation -- you should only need to do this at check-in and then you can use the R.O.C. passport to board the place and the Philippine one in the Philippines. As long as the English name in the R.O.C. passport matches the one in the Philippines passport, and the photos look like you and both passports are valid it should be OK -- it's not exactly an unusual situation. Turn up early as it may take them a while to decide it's OK -- if you're really concerned phone the airline first.

Depending why you have two passports you may want to try and avoid using both of them at any immigration point. There are some situations where it's OK to have both (i.e. if you're a dual citizen by birth) but if you adopted one or other citizenship later you may have been supposed to renounce your original citizenship -- I know this is the case in the Philippines. Therefore immigration may ask about why you have both. However at the immigration points you should only need to show the passport for the country you're entering -- you should only need to show both at check-in for the flight.

As I said, if you want to put your mind at ease you can always call the airline and talk to them. They may also be able to change the name or add the other name to the ticket (although they may charge for this and I don't think you need to do it).