Garran: Flashpoint won't show up as a separate item in your library. It'll be in the DLC list of your Battletech library page (linked next to the version number near the top).
There should be a checkbox for each DLC you have; if it isn't there then Flashpoint hasn't activated for your account for whatever reason.
On my end, at least, the 1.3 and Flashpoint updates occurred together, so if it's there/ticked then launch the game and see if anything about it shows up there. (The actual 'flashpoints' don't appear until you've finished the Arano Restoration campaign, though.)
can you give us a screenshot of what you are seeing in relation to these check marks/DLC list?
I never expected to see it as a seperate library item, but I expected to see it as either a check mark like the Shadowhawk DLC is, and as a backup download copy as well. But I see none of that. So yeah, not activated on my account.