To me this is more of a workflow issue than a technical issue.
In general I would always try to avoid adding a single page to a book or booklet. I almost always regard the spread as a whole. Inserting a single page would shuffle the whole book and ruin the balance of every spread.
If I absolutely had to insert one page I would see if I could get away with adding a whole spread, or perhaps add one page and then another page a few pages ahead to isolate the shuffling to a few pages.
If you are making a book or booklet I would strongly recommend that you work with Facing Pages. You have to see what the reader will see in the finished product. (Working in single pages also makes you lose a tiny slice of your images as you will have an unneeded inner bleed.)
So setting up page numbering is not a problem. Just add Current Page Number markers to your Parent pages and InDesign will automatically handle even and uneven pages.
When inserting a single page in a document with facing Pages, InDesign will actually try to help you maintaining "design integrity" as you call it. Objects will be moved according to the margins. So if you have a layout where all objects are placed in relation to the margins everything moves in good order.
But if you have a layout which is off-center according to the column, InDesign can't help you flip the orientation of the layout (And it shouldn't! It would be extremely annoying having some AI trying to design for me.) It still simply moves objects according to the margins.
If an object crosses the page border it will not be moved according to the margins, but according to the page. This can cause some confusion. Notice in the next example how the magenta and the yellow rectangle change their relative position as the yellow rectangle is moved according to the margins and the magenta rectangle is moved according to the page. Also note the craziness that occurs when the cyan triangle first moves according to the margins because it's inside the page when on an even page, but then ends up exceeding the uneven page and therefore must be moved according to the page when the inserted page is deleted again. That's just how it works.
So one solution to your problem would be to have identical inner and outer margins and not adding bleed before the order of the pages have been approved.
You could still make sure that 3 mm bleed will be easy to add later, no matter if the page ends up as an even or uneven page. For example like this:
Addendum: Achieving automatic bleed using anchored objects
This is a bit of a hacky solution, but you seem to be very interested in being able to move around pages with full page images without being forced to manually adjust the bleed, so check it out and see if it makes sense for you to use.
First create an Object Style for your full page images and enter the Object Style options.
Under Size and Position Options set Size > Adjust to Height & Width and set the Width and Height to the size of a full page image including bleed.
(I have an A5 document here so the width is
148 + 3 = 151 mm and the height is
210 + 3 + 3 = 216 mm.)
Under Anchored Object Options set Position to Custom. Tick on Relative to Spine. Set both Anchored Object > Reference Point and Anchored Position > Reference Point to the middle towards the spine. Set Anchored Position > X Relative To and Anchored Position > Y Relative To to Page Edge.
Optional: Under Frame Fitting Options you can tick on Auto-Fit and set Content Fitting > Fitting to Fill Frame Proportionally.
Now you can simply apply the object style to an image and copy/paste the image into any text frame on a page. Part of the text frame must be inside the page, but besides that the position doesn't matter. The images will be positioned according to the spine, so you can move pages around as you like and the images will always have an outer bleed.
I'm deliberately placing the text frames a bit outside the page so they are easier to grab later. Otherwise they are behind the anchored images.
The reason I'm not just using the Size and Position Options to position the images is that sadly it can't position according to the spine and it also doesn't allow negative position. I hope this feature gets better in future versions.