Carbon Paste is Grease with sand in it, you should never add grease to your Brompton seat post .

just add the grit if you want,
rough up the chrome where you want the seat post plastic shim to grip better.

a few strokes with steel wool around it at that specific spot .

on my Mk2 I replaced the plastic shim .. used Hot melt glue ,

then played the heat gun on the frame, there till the metal
was warm enough to let the seat post squeeze out the glue enough to get the post In.