If you can rent a car with a towbar, then I'm sure you can also rent a trailer that you can return in Stockholm.
Simply put, if you can rent a car with a towbar you can always get a bike rack and transport your bike that way and fit the rest inside the car if you don't have lots of stuff/furniture. So you can bring your bike even if you don't rent a trailer.
You might not be able to book a car with a tow bar or a roof box (if that's the way you want to fit more stuff) for that matter directly on the rental services websites, but you often can if you call them up. Hertz, Europcar etc should offer that service. You can always rent a minibus too if you can manage to fit your stuff inside of a minibus and that should be cheaper than renting a van. You just need to call them up if you need extras such as a tow bar or a roof box. Every rental pickup station might not have them, but you should be able to find a place. You should have no trouble to make it far under your budget here.