The following is my understanding and some of my experience.
I believe Amazon uses something called an ASIN, not an ISBN.
Self-publishing on Amazon does not give you an ISBN, but an ASIN. That kind of number is specific to Amazon. If you wish to have an ISBN on your self-published book (that is, published through KDP on Amazon), you need to assign the ISBN yourself, which you can do. You are also asked through the KDP upload process to identify the publisher--who is YOU, not Amazon. (Unless you have a publisher. To my knowledge, YOU publish on Amazon, and Amazon acts as the print and distribution, and is the virtual bricks and mortar.)
Publishing on Amazon is essentially 'self publishing'--you are the publisher, not Amazon.
Now, Ingram Spark is not a publisher at all. It is also not a 'virtual bricks and mortar,' like Amazon is. Ingram Spark is a print service. And, I suppose, a distribution site, in that bricks and mortar stores can order from them. Ingram Spark does not assign identification numbers of any sort to your book. You do need an ISBN, however. You assign this, and it should not be the number on Amazon for the reasons you describe in your troubles.
If a bricks and mortar store orders from IS, they pay wholesale (roughly half) on the retail sale price (which you set) for your book. This allows them to make a profit when they sell at retail price. NOTE: A bricks and mortar store cannot order from Amazon at wholesale pricing--they must pay the retail price that Amazon, a 'virtual bricks and mortar,' lists (which YOU have set, as the (self) publisher.)
Put more simply, Amazon is a competitor with bricks and mortar; Ingram Spark is not.
Yes, a new ISBN for the Ingram Spark version is a good idea. I believe (but could be wrong) that the Ingram Spark ISBN number will go into their catalog, which Bricks and Mortar stores are very familiar with. If a bookseller wants to order a certain ISBN number, they will know to look in whatever catalog Ingram Spark works with (along with other catalogs, presumably, but they will not look to Amazon in my understanding.).
Back to Amazon: If you add an ISBN to your book before publishing through KDP (Amazon), I think your ISBN does not go into any catalogs. Other sellers will not find it (this is my experience) and wouldn't want to buy it anyway, because it is from one of their direct competitors (Amazon.)
But either ISBN (the one you can add on at Amazon, and the one you enter at Ingram Spark) should ultimately 'lead back to you,' the publisher, assuming you are self publishing. (If you are publishing through a press, then I'd assume they take care of all this stuff.)
I don't know about stickers. ISBNs are supposed to refer to unique editions of the book, and the like, but Cyn sounds like she has direct experience at that level so I'd follow up with her tips there.
The above is my understanding and experience.