You can buy a foldable push scooter for about $100 on Amazon. I have one that I use to go from the parking lot to the indoor track. You should be able to cover 5k in less than 20 minutes without working up a sweat... 15 minutes with a moderate effort. Or you could buy an electric scooter for $300 or so... probably less off of Facebook Marketplace.
If you must go with a backpack, you can use any backpack by adding bungees/elastic bands to connect the two shoulder straps across the chest and the waist. It will stop the bouncing.
EDITED TO ADD: If you get a push or kick scooter, pay a little extra to get the bigger wheels. Bigger wheels are a lot more stable and safer to ride. My scooter has 8"x1 1/4" wheels. I wouldnt go any smaller. Cracks in the sidewalk will catch a small wheel. If storage isn't a problem, you can get a kick bike with 12" wheel in the back and a 26" wheel in front. I have one. You can safely go faster with the bigger wheels. I can average 12 mph at a moderately hard effort on level ground.