[RPG] Does the Monk retain his AC Bonus if he loses his Dex bonus to AC

The Kensai ability says

At 1st level, when a kensai is wielding his chosen weapon, he gains the canny defense ability. This is identical to the duelist prestige class ability of the same name, save that his chosen weapon may be of any type. Source.

The only way that the magus's ability is different from the duelist ability is the way specified, the weapon can be different. Otherwise it is identical, meaning that it would indeed limit the bonus the same way it limits the duelist. The builds you see online are likely overlooking the level restriction.

Interestingly, the ability never states that magus levels function as duelist levels for the purposes of that ability; similar language is very often missing from archetypes, however, and the consensus is that magus levels would count as duelist levels for that shared ability.

Monk's Belt, as you quoted, says:

If the character is not a monk, she gains the AC and unarmed damage of a 5th-level monk. This AC bonus functions just like the monk’s AC bonus.

So, what is the AC of a 5th level monk?

When unarmored and unencumbered, the monk adds her Wisdom bonus (if any) to her AC. In addition, a monk gains a +1 bonus to AC at 5th level.

All of that is "the AC... of a 5th-level monk". Since it functions just like the monk's AC bonus, you will also lose it if you are armored or encumbered.