Compute coordinates of transformed bounding box corners in an image rectified by OpenCV remap

Question

I am computing bounding boxes for objects in an image taken with a 180deg fisheye lens. These bounding boxes will sometimes trigger an event that requires a rectified image. To rectify the image, I am currently using OpenCV's remap with maps precomputed by initUndistortRectifyMap:

map1, map2 = cv2.fisheye.initUndistortRectifyMap(K, D, np.eye(3), P, DIM, cv2.CV_16SC2)
undistorted_img = cv2.remap(img, map1, map2, interpolation=cv2.INTER_LINEAR, borderMode=cv2.BORDER_CONSTANT)

The following are the parameters used:

K = [[169.53432726   0.         322.5714669 ]
     [  0.         169.39413982 289.12455088]
     [  0.           0.           1.        ]]
D = [[ 0.11919361]
     [ 0.29904975]
     [-0.20295709]
     [ 0.05526085]]
P = [[121.41853905   0.         322.57040373]
     [  0.         121.31813842 289.11177491]
     [  0.           0.           1.        ]]
DIM = (640, 640)

Is there a simple, efficient way to compute the coordinates of the bounding box corners in the rectified image (given precomputed maps for x and y)?

1. I believe this unanswered post may have been wondering the same thing.
2. I also think this post was trying to ask a similar question, but it's not clear how to apply the answer to my question (Caveat #1: if the "inverse" map can be used to map a point in the distorted image to a point in the undistorted image, then great--but not clear how to do that from the answer; Caveat #2: it's not clear the inverse map is available for the fisheye module).
3. This post is also relevant--again, not clear to me that it actually answers the question, though
4. I have attempted to invert map1 using the iterative algorithm proposed in this post, but the algorithm does not seem to converge on a useful inverse map. Instead, it returns negative and enormous coordinate values, which are self-evidently incorrect.