Hello, and what a great question. In the conversations I have seen around the use/non-use of white sage and smudging among non-indigenous peoples, the concern is that there is a lack of understanding and respect for the tradition/practice. I personally do not use white sage as I do not feel it is appropriate due to the over-harvesting reasons stated above. As you express the desire to be respectful, I feel that you could use it if you wanted. But as @MeganB suggested, I would reach out to a local tribe to confirm.
I do use smoke cleansing (different from smudging) with rosemary, mugwort, or eucalyptus (which is what it looks like your bundle may be??). You can also cleanse with sound or incense as well.
If you choose not to use the bundle, I strongly encourage you to give it back to the earth rather than simply throwing it away. When I get white sage bundles in one of my witchy subscription boxes, I either offer them to my indigenous friends or bury them.