I imagine a woman in a boardroom pulling a tampon out of her purse by mistake. It would be as embarassing as a guy (same scenario) pulling a condom out of his pocket by mistake. Both initially embarrassing moments, quickly gotten over.
Most women are not embarrassed by tampons or sanitary products. We just don't fancy walking around with one stuck to our shoe, displaying it for all to see, just like I don't like to run out of the bathroom showing all and sundry my toilet paper with the product of my bowel movements. It's a question of taste. I am not embarrassed to go into a store to buy a box of tampons, or condoms, or any product for the nether regions in front of my father, boyfriend, boss or stranger, when I need the item, but I am not in the habit of calling attention to these facts or putting them on show.
I've had my mom find my vibrator once and after two seconds of red-faces I showed her how it worked. I've had a friend pull a box of tampons out of her bag and offer them around in mixed company, thinking they were her pack of cigarettes. Hey you can chuckle about such initially embarrassing moments but they are no big deal.
Your woman friend is not concerned about dropping a tampon and will probably never miss it. There's no need to hang onto it, unless she leaves a box of tampons next time ... that means she's moving in!!!