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The Eiger makes a persuasive backdrop for a bike ride. It does not make the climb shorter. Around Grindelwald, elevation and surface matter as much as distance, so a route that looks brief on a map can still fill a good part of the day.
Start with the official bike route description, then ask a local rental shop about the bike and your experience. Check whether any lift you plan to use carries bicycles on that route and date. Save the dramatic technical trails for riders with the skills. An easier ride with time to stop and look up is a very good mountain day.