Water quality really will depend on water volume. A small amount of water, say 1 liter or less, will tend to evaporate faster than it goes stagnant. I feel that filtration is unnecessary for water features this small.

You could ad some plants to the water, this looks great and will help chemistry a small bit, but given this frog does not spend too much time in the water I would not worry about it too much. Another advantage to plants in the water is they tend to help keep live foods from drowning.

In using a false bottomed tank for this species. I suspect the water would become more of an issue than a help. You may find that setups like these often result in bog-like conditions in small set ups. It easily could become too wet for your frog.