Addressing your various questions:
1) BEH HILIC is uncoated BEH material, I believe. That's a hybrid between silica and silica hydride. It contains fewer silanols and the ones it does have are more isolated and less acidic. Waters has contended that it is uncharged below a pH or around 9, but empirical evidence suggests that in fact it does acquire some negative charge in the same pH range as silica: pH 4. The ease with which an immobilized ionizable group acquires charge is affected by the charge on neighboring groups. Therefore, a titration curve of a material like silica or a BEH material would be a continuum rather than a curve with a sharp inflection point. That accounts for the continuous increase in retention of your basic peptide with pH.
2) Further on that retention: Retention on a HILIC column is a combination of hydrophilic interaction (the partitioning into the immobilized aqueous layer that Gerhard mentioned) plus any electrostatic effects that may be present. In your case, this involves electrostatic attraction between your basic peptide and the (-) charged silanols of the BEH material.
3) Salt shields electrostatic effects in chromatography, both attractive and repulsive. In your case, it shields the electrostatic attraction, which is why retention of your basic peptide decreases as salt increases. What's left is the retention by pure hydrophilic interaction.
4) Basic residues are normally the most hydrophilic residues in peptides. TFA tenaciously forms neutral, somewhat hydrophobic ion pairs with them, rendering them all but invisible to a HILIC column. That's why retention decreased so much. If this bothers you, then use formic acid instead (if you don't need to monitor absorbance at low wavelengths) or a salt. Salts that are quite soluble in HILIC mobile phases and which are transparent at low wavelengths include triethylammonium phosphate and the sodium salt of methylphosphonic acid. Be warned that these exhibit very different selectivity in HILIC of peptides. Phosphate promotes retention of basic peptides and suppresses retention of acidic ones while phosphonate has the opposite effect.
Per HPLC addict: Applying a neutral, hydrophilic coating tends to suppress the charge effects of the stationary phase. BEH Amide is in that category, as are various diol and pentahydroxy materials, our own lovely PolyHYDROXYETHYL A, and so on.
If you'd like to read more on the subject, then try the following papers:
1) Anal. Chem. 80 (2008) 62-76.
2) Anal. Chem. 87 (2015) 4704-4711.