Sadly the Facebook Tablet app seems to be dead. One I miss, along with Comics HD.
When I found my TP after it had been lost for a year, FB started but wouldn't show the news feed correctly. Update alerts came through, but didn't display half the time.
I removed the account from synergies and tried to re-add, but FB complains that it thinks that the credentials are wrong.
You can use m.facebook.com, which provides a mobile enhanced interface. Albeit the WebOS browser isn't the best. I believe LuneOS provides a direct link to this, with the enhanced bundled browser, instead of a dedicated app?
Otherwise install Android (dual boot is straightforward, there are great instructions in the forums here and on XDA developers), Evervolv 4 Jellybean is a stable distro and the facebook app works as per any other Android device, some folks have been able to get Lollipop and Marshmallow to install.
(Incidentally, how did you get your google account to work? I've tried all the patches, openssl updates, certificates etc. but nothing stops the yellow triangle of doom.)