Although movement in 5e is not governed by squares, they are probably a helpful model to this particular exercise. Everything in the universe is effectively measured in 5' increments and using squares to model this will give us an effective answer to this question Let's posit a 50' wall

 x
 x
 x
 x
 x
 x
 x
 x
 x
 x

Now let's look at the veritcal space next to the wall

 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx

Finally we need to have the flat part on top of the wall, and the space above that

  yyyyyyyyy
 yxxxxxxxxx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx

Now, supposing our hero starts at the first y (marked a below), and runs up, he runs out of movement at the 10th y (marked b below).

  yyyyyyyyy
 bxxxxxxxxx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 ax

If you start at a, you might be able to climb up onto the top of the wall. However, if you have a 45 foot wall you are safe. If you have to move to a, a dash would be required to not fall (as you don't have the ability to stay vertical).

The problem here is that if the wall extends at all into the square above the one marked b, then you're in trouble, you have to move into the square above it as diagonal movement is not possible.

This will be both at the discretion of the map makers (if the building extends a bit above the square you're probably not going to move diagonally) and also your DM (he may allow a bit of wiggle room here). Consult your DM before you attempt any 50' wall climbs.

45' wall climbs (and lower) are safe though.

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.