I went from Magwe to Pyay by local bus and later asked a local driver about the possibility of a Pyay-Tangoo trip. He said that the road is a disaster (in various states of disrepair) and you 'd need to hire a private driver with a 4WD to do the trip. I hear there is not much there anyhow, why not head down from Bagan by bus? I stopped in Magwe and spent 2 days & 1 night seeing the sights there, that town was pretty interesting.
Here's my report if you care to read it, photos in link at end
It is possible to get from Taungoo to Pyay in a day by train or bus, via Yangon.
Take an early morning bus from Taungoo to Yangon's main highway bus station, and if you arrive before 2or 3pm there are buses still going. Or if you don't arrive in time, take the 4pm train from the main Yangon station, heading to Bagan, and stopping around 10pm at Shwedagar station just outside Pyay.
This trip does take some time, otherwise you can get a permit and drive across the bad road - ask at Beauty II guesthouse if they can arrange this for you.
You will need a 4WD to get across that "bad road", a regular taxi isn't going to make it. That wil be extra $$$.
The driver that I had from Yangon~Hpa An confirmed the above and also that that route is not particularly interesting. I'd recommend accessign it from the other route (route2), for example form Yangon or coming down from Bagan, Magwe et al. and avoid having to cross through Taungoo (on route 1).