I can't come up with a straightforward way to remove the key waste on the inside. Even using a delicate sawing technique it seems like one could all too easily damage the bottom and/or sides and/or corners.
Sawing may be one of the better methods to remove the majority of the excess, although it does come with some danger a great many operations in woodwork pose a certain amount of risk of doing some damage to the wood (including sanding). This is an aspect of David Pye's "the workmanship of risk".
I was thinking I would probably try doing this first by notching the key with a sharp knife, then by doing some careful work with one or more chisels, but the size of the box may make this impossible unless using skews and furthermore there's always a chance of marring adjacent surfaces when paring flush even in ideal circumstances.
So back to sawing, a quality flush-cutting saw with a flexible blade is ideal for this kind of work. An option is to tape one side of small pull saw, and in a pinch you can also use a hacksaw blade. These two options will leave more wood proud of the surface which will require more effort to clean up.
Once you have removed all or almost all of the excess, you will be left with some cleanup. Sanding into a corner is tricky to say the least. Scraping is a better option in this situation, the work can be tedious but it is what it is.
A small bullnose plane may be the ideal tool to deal with the remainder and make the keys perfectly flush, but not everyone has one of these or can afford one, so careful scraping may be the preferred solution.
I have thought about pre-shaping the key with a corner before glueing in, so that just sanding is needed, but not sure what is best way to go.
I think this is actually a viable option, and actually one of the better ones if you can figure out how to do it accurately and repeatably.
Accurately aligning the cutouts in the keys to the inside of the corner may turn out to be the most critical aspect of this method. Small blocks inside the corners to clamp against (covered in tape or well waxed to protect them from glue) might prove useful.
If you want the box finished inside as well as out pre-finishing the inside of the box* would make cleanup of any glue squeeze-out easier (glue doesn't bond to finished wood). I would use either paste wax or a light coat of shellac for this myself, but spray lacquer could also be perfectly suitable. See this recent Q&A for why you might want to avoid using any oil-based finish here.
If you prefer to leave the inside of the box bare protect adjacent surfaces using tape. Packing tape is great for this as it's wide and very thin, and costs a lot less than equivalent painters' tape.
*If you pre-finish the box pieces before assembly be very careful not to get any finish on joint surfaces.