Ok, if this were my bag, here is what I would try first:
I would definitely give the bag a good long bath, and let it soak. I would let it dry, then start experimenting:
I would try baby powder on one of the spots, to see if I could get it to lighten. If that works, I would do that on all the dark spots... if not:
I would try an oil on the bag, I have used olive oil before , and it darkens the leather, and in this case it might blend this bag so it could look uniform. Try it just on a small non-visible area first and let it thoroughly soak in to see if it might come out the same color. Google olive oil in leather repair, and you can read the pros and cons, I have used it on a bag and I liked the result, I also have used it to darken the handles on a new Louis Vuitton so they would look more aged, and it has been over ten years, those handles are still amazing.. If that didn't work:
I would probably try to re-create the situation that happened to this bag. I would probably be crazy enough to warm the olive oil in one of those butter warmers for Lobster butter, over a candle, and try rubbing some in that way to see if I could get it to match. I would go really carefully, and not-re-cover the already affected areas until I thoroughly tested it and let it dry, I might use q-tips to place the warmed oil all around the stains, without re-covering them.
If none of that stuff worked, I would probably take the bag in to a cobbler and have it dyed, or try to dye it myself. But, since this is grease that has permeated the leather, I would be concerned that dye might not even take evenly, which is why I would consider taking it in.
And, as a LAST resort, if the cobbler gave me no hope, or the dye didn't work, I would get myself a deep fryer, and dip this bag completely, lol!