Nobody said that all the $16000$ people should sit at the table in the same time. So, you build a round table for $81$ people (king Arthur and $80$ of his men).
Have king Arthur and $80$ of his men at the table in rounds of say $13$ minutes plus $2$ minutes to allow everybody from the previous round go away and everybody from the next round to come in and take their seats.
You will need $200$ rounds to have all the $16000$ men to seat with the king ($\frac{16000}{80} = 200$). You can have $4$ rounds per hour. Considering that it must be done in daytime (at night, light from torches and candles won't allow a clear perception of microexpressions), then the king takes $10$ hours of each day with that job, giving $40$ rounds per day. So, in five days, the king would have had everything done.
Consider that everybody has a seat of $50\;cm$. This is tight and somewhat uncomfortable (specially for the king), but except for a few of the most fat soldiers, easily doable. This gives $\frac{0.50 \times 81}{\pi}\;m \approx 12.89\;m$ in diameter. Enough for the king to see the microexpressions of everyone and to have everyone's talk at the table heard by everyone else. You can make the king's seat be larger, with $84.1\;cm$ to make it very comfortable for him and it will be $13\;m$ in diameter ($\frac{0.50 \times 80 + 0.841}{\pi}\;m \approx 13.00\;m$).
Creating a round table with $13$ meters in diameter isn't that hard if you just assemble it from a set of smaller tables, specially if its center is hollow. Among all of that $16000$ men, surely you would have more than enough with woodworking skills to be able to create the parts of the table and assemble it in one or two days.
You could have $12$ or $13$ hours of daylight, but we shouldn't make the schedule too tight to have some space for adjustments and have the king be allow to take a break and have a lunch around midday, so $10$ hours per day is enough. If something goes wrong (like having a bad weather in one of those days), you can simply add one or two days. Since the battle will be in a month, we have enough time.