Use the pill box. Don't worry about it. If you want to do belt and suspenders, take copies of the scripts. I don't even do that. I just have a little card with the names and dosages of the stuff I take. I've been traveleing extensively internationally for 35 years and have never been so much as asked about it.
For those on maintenace meds that they get 90 pills at a time, it's completely impractical to lug along the large bottles, and frankly, nobody expects you to.