Unlike trickle filters, a filter like this would use dissolved oxygen for the aerobic bacteria (like any other kind of filter). It would work fine but lacks the carrying capacity of a wet/dry arrangement due to the lower O2 levels, although adding aeration would help solve this.
That said, unless you're keeping Giant salamanders you don't need the kind of handling rate that a tank full of messy fish need and the elevated flow rates associated with sump filters may be problematic for your axies without careful handling. Oh, you also need an overflow drilled in the tank to avoid flooding too..