This is pretty straightforward. The only person who necessarily gains knowledge on whether or not a creature within the radius of Zone of Truth has been affected by the spell, and therefore is or is not capable of lying, is the caster.
Since the character in question is not considered reliably honest by the others (that's why there's a question of honesty in the first place), the character's own word about whether or not they are affected by the spell is similarly unreliable. If that would work, they might as well not bother with Zone of Truth at all.
This is particularly true if you are hoping for others' views of this character to include a reputation similar to inability to lie. Aes Sedai cannot lie, with magical effects physically preventing them from directly expressing, verbally or in writing, something that they believe is untrue. Trying to use Zone of Truth in that way may even backfire-- without the spell being in effect, people may assume that the character is lying simply because they are eliding the situation in which lying would be impossible to get away with.
The correct solution involving RAW Zone of Truth would be for the character to submit to another character casting the spell, perhaps repeatedly until the character fails the saving throw (voluntarily or otherwise). If your character is really focused on this, building them to be able to produce spell scrolls for Zone of Truth and then giving those scrolls to others so that the spell is as available as possible to other people might be the closest RAW solution involving this spell.
Other options in this area will certainly exist as well, but lie outside of the scope of this question. As quick examples, someone could cast Wish on a character to produce this effect, homebrew spells/curses/feats/magic items can impose it, and so on.
I've run NPCs with inability to lie as a trait, and found mechanics surrounding that to be unimportant (players don't trust them, much like Aes Sedai are not much trusted because they can be misleading without lying). The hard part is actually making sure they never lie, because the players/DM at the table have no constraints on lying whether a character does or not.