Trimming oaks in September or October in Northern Lower Michigan
Roscommon County Michigan
The general rule of thumb for pruning oak in Michigan is as follows: "The highest risk of infection occurs April 15-July 15, but it is prudent to avoid pruning or injuring oak trees until they have lost leaves for the winter, typically from November through mid-March. If you must prune or remove oaks during the risk period, or have a tree that gets damaged, immediately cover wounds with tree-wound paint or latex-based paint."
I'll assume you already are aware of this as you said you have been looking into it and still have questions.
As the hole showing in he photo looks to be a den entry of some kind it's an indication that section of the tree is already hollowing. As the limb overhangs a structure, I think it would be prudent to remove it as soon as is possible.
Risk of oak wilt would be more or less the same from woodpecker damage as it would be from pruning (oozing sap attracts insects which transfer spoors). So although removal will expose tree to infestation at some level it will probably do more to reduce the risk by eliminating woodpecker activity.
As you are past the most critical time to avoid damaging bark you should be OK to prune. Sanitize cutting equipment before uses, leave a wound which is angled to prevent water from pooling up (causes premature rot) and treat wound with some kind of wound or latex paint.
Use caution...oak limbs are very heavy and require some degree of skill to remove safely, especially one as large as it looks to be in the photo.