requesting a table limit increase
is not any better than asking for a private table, both will draw attention. A simple solution is play 2 hands or go to a higher limit table.
Warning the pit(if you will) that you want to increase your max bet beyond table limit is not a good idea.
Even if the pit is not required to call in, does not mean they wont. Of course if a gambler dont care about counter measures or getting barred, then.......
But then again , this is just my little humble opinion of how I would handle the situation.