Greetings
If you rent a bike in Copenhagen, you can take it on any train and there are plenty of places to bike and stay in Zealand. I think for a weekend, you probably want to stick with Zealand as you'll spend a good chunk of your time traveling to go to Jylland.
August - at the leas the first few weeks - can be very busy with Danes on vacation, so campsites/hostels may be less secluded at that point and you should definitely make reservations ahead of time.
I know that there are a series of fairly inexpensive and simple campsites and hostels which are placed at intervals suitable for cyclists. I would do a google search for Denmark and cyclists, as there's a national bike organization that has all this info and also puts out good maps with everything a cyclist needs to know. The Danish tourist website has a good page on cycling which has lots of links:
http://www.visitdenmark.com/uk/en-gb/menu/turist/inspiration/aktivferie/cykel/cycling.htm