Go for it. The scholarship difference makes it worth it. Work specifically on math and science. Just improve those by one point and you’re good. 
“First, Middlebury is not test optional.”
It is as far as the ACT/SAT is concerned, which is what this student cares about. She can submit SAT subject tests instead.
“Second, a 31 would not help at Bowdoin and similar schools given the middle 50% was 30 - 33 with an average of 32.”
I disagree. That’s the average of admitted students, not applicants. If you fall in the middle 50% of admitted students, you can be certain that you’re outperforming a lot of the kids who got denied. That is doubly true where, as here, only students who think their scores will be helpful are even submitting them to Bowdoin. Anything that puts you close to the average (or above) of admitted students will make your application stronger. You fit the profile of an admit.
Let’s say the average 40-yard dash time for an NFL player is 4.8. If you’ve got a 4.9, that’s going to help you get drafted. Being just below the average of a successful player means you’re way ahead of other hopefuls. It would be a big mistake to hide that stat, and a big mistake not to send a 31 to Bowdoin.
Yes, I would say go for it and test again. My son had taken it twice ( the first time a 31 and 2nd time a 34) and the scholarship opportunities were amazing. There are alot of schools that will pay your tuition for all 4 years with a 34 but he also has a GPA of 3.93 and in the top 10% in his class.
If your interested in particpating in athletics and are good enough, I think checking outD2/ D3 schools would be worth it too.
Good luck!