The first place to look at may be your webmap's JSON : http://www.arcgis.com/sharing/content/items/178fc074fc784f5cadd5eba053e73b30/data?f=pjson
Search for "id" and you'll see the ids assigned to map services and feature layers that you added explicitely. For example : IOT_234.
If adressing a layer inside a map service, add _X where X is the layer's id within the service. For example : IOT_234_0
Your particular case has :
IOT_1553 : bridge replacement zoom out
IOT_1553_8354 : bridge replacement zoom in
IOT_1318 : bridge rehabilitation zoom out
IOT_1318_5629 : bridge rehabilitation zoom in
IOT_234 : improve our Tulsa
IOT_234_0 : major street labels
IOT_234_1 : project details
...
IOT_234_7...
Your best bet may be to store the exclude list within a new property inside the layer list widget's config file. Then you patch where I said, checking whether the id belongs to the configured list.