So how are you thinking this would work exactly? For items in FBA AMZ has a choice for a "gift bag" for many items. You want something "branded" as what, YOUR company name? Or the BRAND you are selling? How would AMZ do this? They would have to stock the bags as a separate item, and then pull them as customers ask for them, that would never happen for $3, not to mention the brand would probably want more than that for their branded bag.

If you want to do this, you would need to create a totally new sku for the item WITH an included gift bag for $3 more or whatever and include it with that sku. A business I had here in the past, we would add things to differentiate us from the competition but it had to be it's own ASIN. They were often $5 or less items but created a sense of added value to the customer as they were a small accessory that they would have had to buy anyway like a cable or adapter. We charged the same amount everyone else did but you got a little freebie with it, and since the amount was so small, even our mostly mapped product were OK with it.

AMZ has NOTHING but problems now with the bots running wild, DD+7, reviews being removed in bulk etc. All I am saying is if you think it's a great idea then you could figure out how to do it w/o AMZ involvement, just create a new ASIN and include it