Catalysis of the reaction: 5-(carboxymethyl)uridine34 in tRNA + S-adenosyl-L-methionine = 5-(2-methoxy-2-oxoethyl)uridine34 in tRNA + S-adenosyl-L-homocysteine.
tRNA (5-carboxymethyluridine(34)-5-O)-methyltransferase activity