Need help to create a selection and then remove background from 100+ images

NB: I did read a few other answers to similar questions, but none were sufficient for my task.

I have a series of 100+ images I created for a client (see example) which are 360º images, transformed into a horizontal cross shape (using Flaming Pear software). Client is an architect and will be creating a book of reference images of a building that requires renovation. The shape was chosen so that text and other photos could be added in the white space around the main image. This allows them to highlight problem areas etc. The problem is that the "white" background areas are not uniformly white (instead they range from white to grey). which messes up the look of the book (they are using Word). They are now asking if the images can be supplied with no background (make it transparent?) so they can put into a Word document and then add other elements around the image. I hope this makes sense. I understand I should be using Actions in conjunction with Batching, but my knowledge is very limited. I have made a note of the exact pixels from which the background area radiates i.e. the corners of the central square of the image. I want to remove the background from those points on every image.

How do I accurately make a selection of the grey background using the pixel information I have? NB: I tried various auto-selections and it selected parts of the image, so would like to make manual selections. Do I then save this as a mask to be used on the other images?

What steps do I need to achieve this in an efficient way when recording my Action? e.g. apply selection/mask, cut background, flatten, rename, save in original folder.

When saving the new image, how do I append to the original filename, so that the can import the new images alongside the old, but identify the new ones easily e.g. FileName_removed

Many thanks for any help.

Gary