How can the double be detected?
There are three main classes of illusion spells and then a handful that are relatively unique. The first class is spells that exist only in the mind of a target; they have no external form (e.g., phantasmal force). The second are spells that have a tactile component, and are solid enough to be touched and support weight (e.g., phantom steed). The third class of illusion spells are external images - things that appear present to all observers, but which are not tangible and which solid objects and creatures will 'pass through'. Mislead is in this third class of spells; the duplicate is an image that can be seen by all but which begins completely overlapping with your body and can later pass through other things.
Spells of this third class tell you that they can be detected by 'physical examination'. Minor illusion, disguise self, silent image, major image, hallucinatory terrain, seeming, programmed illusion, and project image all contain some variation on the phrase "Physical interaction with the image reveals it to be an illusion, because things can pass through it. A creature that uses its action to examine the image can determine that it is an illusion with a successful Intelligence (Investigation) check". All of these spells work this way except mislead. Mislead doesn't tell us how it can be detected.
A foolish application of 'spells do only what they say they do' might conclude, 'well, I guess there is no way to detect that the double is illusory, since it doesn't say that there is.' You can rule that way, and then the answer to this question is that of course the illusory bard can perform, because it is not possible to detect any difference between them and the real bard. A less credulous response notes that the description of mislead appears to be missing the paragraph on detection and reasons that it likely was supposed to resemble all the other illusion spells of its kind. Personally, I find it likely that mislead is supposed to be detected like project image, given their other similarities. Of course, a DM could decide for themselves what are the criteria for detecting mislead - in which case that decision will then inform their own answer to this question.
Physical interaction with the image reveals it to be an illusion
Let's assume for the rest of this answer that the illusion can be detected by physical examination - and only by physical examination. That is, it is such a convincing illusion that anyone seeing or hearing it would believe that it is the real bard. Only touch will reveal its falsity. The spell says you can "make it gesture, speak, and behave in whatever way you choose." So you know you can make it speak, but can you make it sing? Can you make it play the lute? Well yes, because it is undetectable as an illusion by sound. Only physical inspection reveals the illusion. So not only does the illusory lute sound believable, but the weightless image will nonetheless creak the floorboards as it strides the stage. In a quiet room, the audience will hear the rustle of its clothes as its walks by. Its auditory output is completely convincing.
So long as the bard's double's performance doesn't involve touch, it will be convincing. It will provide an excellent opportunity for you to 'snoop around' - leave the illusory bard singing songs and playing an instrument to distract, while you sneak off invisibly to investigate. One caveat, though - remember that, per the spell's description, you can only use either your own senses or those of the double, not both at once. If you slip out of the room, you are blind and deaf as to what is going on with the double - you certainly can't have it moving through the crowd. It may be fine while it is up on the stage, working through your set - but what happens when a fan throws a flower - and it passes right through? Or when the lord commands, "Enough ballads, let's have something the people can dance to!" and the double goes on playing regardless?
So what do you mean by perform?
There is, of course, the Performance Skill:
Your Charisma (Performance) check determines how well you can delight an audience with music, dance, acting, storytelling, or some other form of entertainment.
So long as your entertainment doesn't involve touch (avoid paired juggling!), your double should have no problems with this.
In my experience, bards often choose the Entertainer background, giving them the feature By popular demand:
You can always find a place to perform, usually in an inn or tavern but possibly with a circus, at a theater, or even in a noble's court. At such a place, you receive free lodging and food of a modest or comfortable standard (depending on the quality of the establishment), as long as you perform each night. In addition, your performance makes you something of a local figure. When strangers recognize you in a town where you have performed, they typically take a liking to you.
Again, so long as your performance requires no physical interactions, you should be fine having the double do this while you are elsewhere.
For a bard, though, 'performance' might also include two of their class features:
Song of Rest:
Beginning at 2nd level, you can use soothing music or oration to help revitalize your wounded allies during a short rest. If you or any friendly creatures who can hear your performance regain hit points at the end of the short rest by spending one or more Hit Dice, each of those creatures regains an extra 1d6 hit points.
Countercharm:
At 6th level, you gain the ability to use musical notes or words of power to disrupt mind-influencing effects. As an action, you can start a performance that lasts until the end of your next turn. During that time, you and any friendly creatures within 30 feet of you have advantage on saving throws against being frightened or charmed. A creature must be able to hear you to gain this benefit. The performance ends early if you are incapacitated or silenced or if you voluntarily end it (no action required).
Here it is a bit murkier. Neither of these powers meets the definition of magical - they are class features, not magical abilities per se, and they don't even require your concentration. As written, they just rely on you being able to be heard - which the double can be. RAW, I think you should be able to use them through your double, but I would certainly understand any DM who drew a line here and said that an illusion would not do in this case, the actual bard needed to be present to power the abilities.