Well, after much experimentation, I can tell you what didn't work, and then I will tell you what did work!!
Didn't Work (to save you the hassle):
91% rubbing alcohol
Tide Stain Pen
Huggies baby wipes
Lysol Kitchen Cleaner
Acetone (and don't try this, it bubbled up a tiny dot of the PVC coating on my bag!!)
the scum buster recipe involving white vinegar and dish detergent mixed together
coach leather cleaner
coach fabric cleaner
Meguiar's Ultimate Compound
Dryel Stain Removing Spray
acrylic paint (impossible to match it, and the PVC coating is so thick that it casts a shadow... not a good look, but a little soap and water and you can remove it easily, thankfully!)
So... after many depressingly failed attempts, the bag was still ruined!! I wanted to cry... but instead I figured out what the bag was made of from a post that said it is safe to use bleach on the white part of the bag to remove stains. I figured out that it is PVC, and highly porous. I then googled PVC cleaning, and found some British website (no idea where, it was late last night), that had detailed instructions for removing stains from both PVC tablecloths and PVC flooring.... the recommended thing for tablecloths was some mold removing spray only available in the UK... I don't live in the UK, and Amazon UK doesn't ship the product to the US, and plus I am not that patient, so that option was out!
Ok, so I then read the instructions for removing stains from PVC vinyl flooring... and viola!!! Success!!! This is an overnight kind of thing. I am not responsible if you destroy your $700 bag following my instructions!! I will show you photos of the process, and I will tell you it worked on this particular Burberry Nova Check PVC and Patent Leather bag. Any use on any other bag is strictly up to you and you are responsible for the outcome!!!
Ok, so, the instructions for flooring said to take bleach and wet a piece of white sheet and leave it on the stain overnight.
This Works (HOORAY!!!)
I used white cotton makeup pads, soaked them in regular Clorox bleach, and cut them to the size of the stains, then laid them on. I did this on the entire bag where the stains were, including where I damaged it with the acetone... the bleach did no damage to the bag, did not discolor the plaid in any way, and removed almost every bit of each and every stain. I thought the bleach had damaged a weakened area of the PVC on another part of the bag, but I am now convinced that that damage pre-existed the bleach and simply became more visible when the ink was removed from on top of it.
This process, of soaking the cotton in bleach and laying it on the stains, took over 12 hours to fully remove all of the ink from the bag. This bag was really badly stained, no telling how long the stains had been allowed to soak in on this bag, and yet, they are gone!!! I am guessing that most stains when treated in a timely manner won't take nearly this long, so please make sure you check the progress often to be certain the bleach isn't harming your bag... After a few hours, the cotton pads degraded into a cotton pulp... I just kept laying this on the bag on the stains that still were visible, until one by one, they all disappeared!
I have to say, I am a fan of Burberry PVC bags now... they can take a lot of abuse!! They are also pretty easy to make look new again... there are still some faint beige marks on the bag where some of the ink was. I think those will go away if I leave the bleach on longer, but I was getting impatient!!
cotton pads on dipped in bleach and laid over the stains
sideways view of the bag right after I removed the cotton (the white stuff is disintegrated bits of the cotton pads)
purple ink stain from some kind of magic marker, this is halfway through the treatment, it looked black when I started
desintigrated cotton pads soaking in bleach, mine had little scrubby things on one side which is what those white dots are
cotton with bleach on it sitting on some of the stains
view of the bag after stain removal... I was grinning from ear to ear when I took this!!!
The other side of the bag after stain removal... both sides of the bag were stained
light beige mark leftover where there was once a dark ink stain, I think further bleaching will remove it completely
The front of the bag where you can still see (if you really look hard) light beige traces of the original stains, but mostly gone!!!
the imperfection that was discovered after all that ink was removed... this was the area that looked like a cube in the before photos
A closer look at that imperfection... considering the bag was so cheap, I am perfectly content with that!!!