Post-2002, it is no longer possible to obtain a Republic of China passport and exercise your status as a Republic of China national without household registration
by virtue of your Chinese ethnicity. You would either have to show that either parent has/or had during your lifetime a ROC passport/ID card or that you were born within the official borders of the Republic of China. How did you dad renounce his Chinese nationality, and was this done before you were born? You basically have to be born to a national to be a national, but if he was no longer a ROC national when you were born, then I don't see how you could be a ROC national.
Either way, being an unregistered national doesn't give you residency rights in Taiwan, and you would still need a permit to work if you concurrently hold Philippines nationality. The only thing it does is entitle you to a once-extendable 3-month entry permit which allows you to stay in Taiwan for 6 months. Even if you are considered a ROC national, you would need to legally reside in Taiwan for a lengthy period of time before becoming eligible for a National ID card and other benefits.