How to separate peptide on SDS PAGE?
Image is of an 18% Tris-tricine small-peptide SDS-polyacrylamide gel stained with coomassie blue dye [source].

Your peptide is 1.6 kDa; it would not run off in 20% gel if you don't run it for a long time. Also, try to prevent heating up of the gel (you can run it in a cold room or in front of an A/C vent).  However, it is better to use other techniques such as chromatography. You can also try dialysis if other unwanted peptides are much larger.
Protocol for Tris-Tricine SDS-PAGE.