Although bedside table is familiar to me, I tried bedside cabinet (the term I usually use) - but this was rejected. However, an internet search of retailers uncovered a roughly 50-50 split in naming this piece of furniture, with no discernible design difference. Bedside tables were just as likely as bedside cabinets to include a drawer and/or a cavity with a top-to-bottom door. The manufacturers clearly see no distinction between the terms. On that basis, I think Duo should accept bedside cabinet.
Anyway - what is a chevet? Apparently, it's a "bedhead".
The word has also been taken into architectural or ecclesiastical English to mean "the apsidal termination of the east end of a church". No, I'm none the wiser either... (Oxford Dictionary)