The tenant claimed he was injured in the process of the work.  

That lawsuit is a claim for fraud, and that’s what fraud typically is...a misunderstanding and someone being “injured” and wanting to hold the other responsible for it.

If someone injures themselves and sues, they will be suing the LLC and not you personally.

As a public official I was once sued with a raft of others in a wrongful death suit when one of our EMTs hit a car on a run and killed a young girl and injured her father.

This really injures the "buy now and you may be able to refi soon" concept.

Would love to hear suggestions for providers that work well with STR managers, and if you think General Liability Insurance is enough to protect against lawsuits in the event a guest gets injured at a property I manage, or damage occurs to their personal property as a result of a leak, fire, etc..

If someone injures themselves and sues, they will be suing the LLC and not you personally.

If it is mandatory and if we don't install it and if our tenant got injured by fire, then I think it can be big problem. 

Also realize the liability you create for yourself if a tenant or their guests get injured by your half way handywork, regardless if it happens to be plumbing, electrical, HVAC, or carpentry.

@Helena Wu a Premises-only Umbrella policy protects you from lawsuits from anyone on your property that is injured and blames you.