If this is a major story beat in the campaign, however, you might consider running it as a 4e style skill challenge. Set a DC based on how difficult, in general terms, you think it will be to influence the royal court (see the Conversation Reaction table, DMG pg. 245) and ask the party members what skills they want to use to earn the favor of the throne. Obviously the bard can use dex (lute) and cha (perform,) but the cleric can also use wis (religion) to put the party's requests in a context favorable to the kingdom's divine patron, the wizard can use history or investigation to point out the ways in which favoring the party is good for the kingdom generally or the king and courtiers specifically, and one or more insight checks can provide advantage to the other skill checks. You can determine the end result based on the number of successful skill checks the party gets before 3 failures: less than 3 gets them kicked out; 3 - 5 get them begrudging acceptance, or perhaps support from the king with opposition from several courtiers; 6 - 8 gets them what they asked for with some measure of enthusiasm; 9+ gets them a royal charter and free access to the castle armory.
Skill challenge is a great idea!