Renting bikes each place visited in Denmark or in 1 place and carry them, and how will they be equipped?

We are staying 7-10 days in and around Copenhagen in August. We plan to tour the city by bike (by ourselves) and take trains out for day/multiday bicycle trips in several places.

My question is whether we should try to rent the bikes in Copenhagen and carry them with us in the train, or rent at the station where we arrive. This is both in terms of price (will we get a much better fare for a 9-day rental rather than three 3-day rentals?) and also in terms of the equipment of the bikes: maybe it's better to have city bikes in Copenhagen and mountain bikes in, e.g., Møn?

Also, for our 3-day tour to Møn at least, we need rear racks that'll accept (Vaude™) panniers, I think they're pretty universal but the rentals I get in France are usually not compatible. Any info whether you can get that appreciated.

Some more details if you want to know/have further thoughts on feasibility:

For Møn, we plan a 2 or 3 days bike tour: leave our luggage in Copenhagen with LuggaeHero, take a train to Vordingborg, stay first night in Stege after going to the beach North of it, then following day to Møns Klint and back to Stege, or elsewhere, and then a smaller tour in Falster maybe before we get back to catch a train. We're not sure whether we rent the bikes/keep bikes rented for a longer period from Copenhagen or whether we rent bikes in Vordingborg.

Then for Aerø we can't leave luggage in Copenhagen or carry bikes from there as we are not going back to Copenhagen after that. We plan to get train to Odense, leave luggage in left-luggage service at station, rent bikes and cycle or take train to Svendborg and ferry to island. Back next day, via Egeskov probably? Again, not sure whether we cycle Fyn or not, and if not, whether it's best to rent bikes somewhere in Fyn or direct in Aerø (which means we need to use same port there when arriving and leaving). (not e-bikes preferably)