Punta Cana and Puerto Plata are 3 hours drive from the capital so that more or less excludes a day trip to Bayaguana. There is a back route from Punta Cana that would avoid the capital but it would still take too long IMO, especially on public transport.
So yes, you would have to go to the capital and get a guagua (local minibus). Bayaguana is not that far from the capital but it is at the same time 'remote' - the road connection is poor and it is not served by either of the major bus companies (Caribe or MetroBus). The guaguas leave from Parque Enriquillo and the trip takes an hour or two - depending on how many stops the bus decides to make.
If you don't want to drive (wise choice

) another option depending on budget would be to hire a car + driver for the day for a pre-arranged fee.