Any idea as to the species?
and exact positioning of cavity and what has caused the problem?
It will be very hard for any formal test to be devised for such a situation as timber is can vary massively from tree to tree depending upon knots and other things. most timber used in construction is still graded by eye as no machine can accurately assess such things as grain, proportions of early/late growth.
The other issue is that a cavity in the tree will accumulate debris that will trap moisture and enable fungi to grow only worsening the issue.

I would accept the advice of the profesionals. As you say "the tree is huge" and has what can only be defined as a weak spot at the base. The wind loading on a large tree is quite large and then coupled with the lever action at the base could cause the tree to fail, and a hundred year old tree falling over could cause a lot of damage.