According to the wiki article on bridges it is impossible to remove those floors tiles while the bridge is constructed. The simplest solution is to deconstruct the bridge, dig out the floor tiles and then reconstruct it. This is a bit of a pain, and has a few other considerations you will want to be aware of.
You will want to ensure that pathing still works while you are doing the remodel. Since you have already built the rest of the moat I would just make another temporary bridge with no controls and then remove it when I was done.
Managing the mechanisms involved is also a minor concern. From your question you have hooked the bridge up to a lever. When you deconstruct the bridge you will get the mechanism in the bridge back, but the mechanism used in the lever to connect it to the bridge will stay part of the lever, unused. You have two options, leave it there and connect using another mechanism, or deconstruct the lever (which will disconnect it from anything else it is connected to potentially leaving dangling unused mechanisms in them, which could be recovered by deconstructing and rebuilding all of those buildings) and rebuild it. In either case you will need to reconnect the lever to the bridge.
The advantage to leaving the mechanism in the lever is that it makes the resulting lever more valuable, which can produce a happy thought in passing dwarves (I will often put my main gate lever in my main dining hall for just this reason). However if your lever is out of the way this doesn't help much. Still mechanisms are only one stone.
The advantage to reconstructing the lever is two fold. Firstly you get the mechanism back. How valuable that is to you is something that depends on your fortress' economics, but in most cases is not a big concern. Secondly it preserves the mechanism/connection ratio which can make figuring out what you did easier in the future. For example I often find myself asking, "Did I remember to hook that lever up to the new gate? well let's see, there are three gates and four mechanisms, so I must have."
The short version is, you can't dig it out normally, but you can take down the bridge, dig it out and then rebuild the bridge.