How can we make a good business case to convince his manager to let him participate in a project for which he will have to be out of office for two working days?
As you stated, the person concerned is already looking for new challenges - this is not likely to change and there are really two options here, as far as the company is concerned:
- New challenges are found in the company for this person.
- New challenges are not found in the company for this person.
I will start with analysis of the second case - if this person cannot find a new challenge in the company, they will certainly start looking elsewhere. They are unsatisfied with their current work (otherwise why look for new challenges) and are looking for a change - if this will not happen within the company, it will be outside the company and the manager in question is left without them.
In the first case, there are several options as well:
- The person concerned moves to a different department (perhaps some other sympathetic manager will take them on) - this is good for the company, as this is someone who knows and understands the company and customers already. It is also good for the person - new challenges. Not so good for the manager in question, as they are left without this person.
- The person concerned is given new challenges taking them part time away from their current job. Good for the person and company. The manager still has some of this persons' time - they can train someone else, for example.
When looked at this way, the best way forward is to let this person go for a few days a week.
The manager himself should be concerned about relying on a single person - this is not good management and not a good position to be in. As a manager, this person need to ensure that the metrics and SLAs they are responsible are met - whether the specific subordinate is around or not. Having this person train others, document their roll or be available for an occasional consultation are all ways to manage this, while letting this subordinate grow.