Is this possible, or are bikes on these trains not permitted?
I've just had a quick look on the cp.pt site, which has an English translation, and suddenly CP are being a lot more reasonable about bicycles on the Intercidade trains than in the past (it used to be nao). They now say:
Bikes can be taken on the Alfa Pendular and Intercidades trains so long as they have the wheels removed and are packed as luggage that is no larger than the individual luggage space per passenger in the luggage racks or under the seats.
(Alfa Pendular are special fast trains that run Faro-Lisbon-Porto)
So you could take an Intercidade train to Evora or Castelo Branco - or somewhere in that general area - from Lisbon, if you can pack your bike as above.
Their network diagram also shows the possibility of a Regional train from Lisbon to Castelo Branco where, depending upon the specific rolling stock, you may just be able to wheel it into a luggage wagon. But these tend to be very thin on the ground, as well as being very slow.
Unfortunately definitive statements on wheel-on bicycle service on Regional trains in Portugal is only available from local sources in the railway itself.