Liability goes further than a venue owner if they don't have the right license (it gets confusing quickly). In general, one can get such licenses through national organizations like ASCAP and BMI, and continental ones like SESAC a lot of times indirect of the original rights holders for recording. Rights royalties and performance rights are different concepts, and you can can one, but not a license for the other and have issues otherwise at a venue, even if you wrote the work in question if you are a member of the above organizations.

I don't make money as a performing artist though, just as a visual one, so I don't have case experience beyond differentiating satire and parody in visual work. In general, parody has more protections from a fair use standpoint than satire. In my work, as with many others, the line can be interwoven from a layman's view, but under the law it is argued as one or the other.

In trademark law though, if you are arguing in the US under the Lanham act, damages are only compensatory for the actual loss without willful infringement. This means that a trademark holder can only recover actual damages incurred, no punitive damages or attorneys fees without proving certain things. Even if a company successfully argued that a work on the fence between parody and satire was just satire and not entitled to fair use protection, they would mostly likely be limited to actual damages, which in many cases is actually calculated as zero. That's the ELI5 of that, but of course it is immensely complex, and subject to new arguments and clever lawyering on both sides of the deal.

It's best to keep discussions with opposing counsel in any alleged breech civil, and to consult a lawyer as not to risk punitive damages when dealing with any trademark case. Most of the time you are not worth it to them, and if you are, you probably can afford a good lawyer.

When it comes to copyright law though, all bets are off. If your work is not in anyway transformative, you are in for rough days. There are statutory damages for registered copyrights, and can be punitive damages even when it is not. I would always be consulting a lawyer in the music business for anything regarding cover work, or liability in dealing with a venue, even if I were just supplying the equipment. Lawsuits are broad usually to name the defendants who have the most money, even if their negligence in the matter is close to nil.