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. [protein]-L-isoaspartate + S-adenosyl-L-methionine = [protein]-L-isoaspartate alpha-methyl ester + S-adenosyl-L-homocysteine Belongs to the methyltransferase superfamily. L-isoaspartyl/D-aspartyl protein methyltransferase family.