Here's another option:
You could think about taking the little green train from Badajoz to Entroncamento, with lots of connections to Lisbon--it runs twice a day, and you'd have no problem having a bike on that--the train isn't that heavily used...we know it well because it runs past our place. Recently upgraded to 2 runs a day, from 1. The little green train is two self-propelled cars that just do the mostly south-of-the-Tejo run from Entroncamento through to Badajoz. It's still the only Lisboa-Madrid rail connection possible, since they shut down the night train between the capitals at the beginning of the Covid restrictions.
(For others, this is a neat little train trip; you can visit some of the towns in the mid-section of the Nascente as documented by
@jungleboy. )
Update: it looks as if a third option is possible, transferring at Abrantes rather than Entroncamento, as has been the usual. If youtook the evening train, the transfer in Abrantes would likely be simpler; it's a smaller station than Entroncamento and you'd get your "into Lisboa" space sooner. As you can see from the schedule screenshot below, it takes an hour longer because you're staying on regional trains, but it might be worth it.
Note: re prices shown, don't forget, if you're over 65, it's half-price.
View attachment 143331
At CP - Comboios de Portugal, you can check train times and discounts and buy tickets online. Further details.
p.s. OR, if you were going to "cycle the works" originally, you could take this train as far as Barquinha, just one stop short of Entroncamento, and then you could "reverse-cycle" the Barquinha >> Lisboa section of the central Portuguese caminho...