I wrote the script [Userscript] Wanikani Item Inspector that may be able to do what you look for. This will require some configuration though. Here is the procedure. After installing the script you go to the settings.
In the settings you go to the Tables tab and create a new table using the New button. A table titled <untitled> will show up. Rename it to Burned Items or whatever you want using the Edit Table Name field. Don’t delete the other tables because they may eventually come handy.
Once this is done you make the new Burned Item table the default that will display when Wanikani is started or refreshed. Move the table in the top position using the up arrow. The table in top position is automatically the default.
The you need to configure a filter to select the burned items. Make sure the Burned items table is selected and go to the Filters tab. Click the checkbox for SRS Level and select Burned. This will fill the table with your burned items.
There is no way to filter only the recent burned items and I can’t implement one because the filters come from the WKOF script which is not written by me. There is a workaround. Go to the Contents tab and and change the Primary Sort Criterion to Burned Date. This will cause the recently burned items to show up first in the table. Those that interest you will show up at the beginning of the table and you may easily ignore the others.
You will find in the Contents tab options to select the information that will be displayed in the table and tooltips. Configure this as you see fit. You should display the Burned Date somewhere to track when the table displays older burned items you want to ignore.
You may want to try the Random Selection feature. It picks a random number of items to display in the table for studying. Every time the table is refreshed a new selection will show up. This may be used to pick a selection of burned items (no date limitations) for studying. Every study session will use a different selection. In the default configuration Random Selection is disabled and all items are displayed.