.... because Verin's plan A was to free herself of her Oaths to Shai'tan by using the Binder, which implies the reason why the BA cannot betray their oaths is because they've been sworn on the binder, not for any other reason.

Ingtar renounced Shai'tan and betrayed his oaths, to help Rand. Nothing stopped him.

Of course, it would be quite possible to make something very much like the Binder for a non-channeler, you simply have to compel him.

An important point to remember is that few, very few DF swear their Oaths to Shai'tan anywhere near SG. Swearing to him anywhere most likely involve attracting his attention on you, so he knows you're his servant and all, but it's terribly unlikely the True Power or OP is involved in any way, except for the BA. Even the Forsaken can betray their Oaths to Shai'tan, as we've seen with Asmodean.

The bound oaths are almost certainly unique to the BA, making them the most constrained servants of the DO after Shadowspawn. To preserve their ageless look, Ishamael had no choice to introduce a triad of oaths on the Binder to replace the WT's. He or his master chose to include an Oath to make them incapable of betraying Shai'tan. The BA are powerful servants, Shai'tan's best before the Forsaken returned, and Ishamael was absent for long stretches during the Age, so constraining the BA was an opportunity the Shadow took, because it could.