I am getting so frustrated. First my motion capture performance doesn't get captured in the sequencer via Live Link (even after flattening the channel), then even expressions using the face puppet expression tool don't get captured. Is there something you're meant to do that I'm just missing somehow?

Are you using CC3 or Meta Humans?

CC3+ 

Tiny parts of the animations carry over, but it's all screwy, take a look:

https://youtu.be/S4cbNmMPbRk

Yes , I know what you mean.
IF you don't get good LL fps , better export your animations via fbx , import to UE4 and swap the LL recorded animations with the FBX one in sequencer.
Also try using moveable lights , guess you are using static lights that have to be backed , you have to tick "build" above in the menu after every change you made when using static lights (game engine stuff for better fps performance). If you want the best quality , use "Moveable Lights" !
Lights are most important in UE4.