Catalyzes the methyl esterification of L-isoaspartyl residues in peptides and proteins that result from spontaneous decomposition of normal L-aspartyl and L-asparaginyl residues. It plays a role in the repair and/or degradation of damaged proteins (By similarity). [protein]-L-isoaspartate + S-adenosyl-L-methionine = [protein]-L-isoaspartate alpha-methyl ester + S-adenosyl-L-homocysteine Monomer. Belongs to the methyltransferase superfamily. L-isoaspartyl/D-aspartyl protein methyltransferase family. Glu-51 is present instead of the conserved Ser which is expected to be the active site residue.