I don't know that anyone will be able to help you with such a specific question. Your best bet would be to email some rental shops in Gothenburg and ask them. Unless there's a chain of related shops in both cities - and Google doesn't encourage me to think that there is - I suspect the answer is no.
An alternative, if you are flying in and out of the same airport, would be to rent a bike in your airport city and transport it by train to and from your endpoints. Danish trains are pretty good about such things, though it looks like Swedish trains may not share their bike-friendliness (unless you rent a folding bike, or are prepared to break it down and pack it in a box).