LH (similar to all other airlines) has no legal/contractual obligation to check your luggage if you're "connecting" on a different ticket. Their CI agents very often do so, though, as a courtesy. Asking them to grant you a higher allowance for the second flight because your first flight is business is stretching the agent's willingness to go beyond rules to help you. In other words, you can't count on that, it will fully depend on the agent's "mood".