Your observations are not surprising. Electrostatic effects don't disappear just because you are superimposing hydrophilic interaction on top of them. Phenomenex' Kinetex material is a core-shell version of uncoated silica. Uncoated silica is covered with silanol groups. Silanols have a pKa around 4, which means that uncoated silica is a cation-exchange material at any pH above 3. In order to elute positively charged analytes (cf. your peptides) from a cation-exchange material, you need salt in the mobile phase. How much salt? Read my paper that introduced ERLIC (Electrostatic Repulsion-Hydrophilic Interaction Chromatography), cf. the following link:
http://pubs.acs.org/doi/pdf/10.1021/ac070997pThis paper goes into considerable detail on the effect of salt when HILIC of charged solutes is performed on charged columns. Briefly, though, you can suppress most electrostatic effects with a minimum of 20 mM salt in the mobile phase (cf. your own observations). That's 20 mM overall, not just in the aqueous portion of the mobile phase. That amount suffices to form an electrical double layer of counterions on top of the charged surface, shielding electrostatic effects. For particularly highly charged solutes, use 40 mM. For extreme cases, such as aminoglycoside antibiotics, you may need as much as 120 mM salt in the mobile phase in order to get symmetrical peaks.
HILIC materials with a neutral coating have less of a problem with charged analytes than does uncoated silica. I recommend one such material for an application like yours. However, you will still need some salt in the mobile phase. What mobile phase did you initially try using, and what are the details of that gradient that you're using now (i.e., composition of mobile phases A and B and gradient schedule)?