This was not an involuntary denial of boarding. You may satisfy yourself of this by looking at 14 CFR 250 and its following sub-sections. It also does not matter whether your daughter's booking was linked to your or not. Presuming that she was too young to fly alone other than as an unaccompanied minor, she would have had to check in with an agent and a responsible adult (you) no matter what entries were made.
None of that excuses what happened.
I would send UA a clear and concise note which contains only the detail specifically relevant to what happened. The reason for needing agent assistance is irrelevant. But, be specific as to language and as to her actions, e.g., jumping on the scales. Include a detailed list of your expenses (with receipts) as well as a note about the missed night. I would read it again and see if there is someone else who can review it to make certain that it is short, tight, but omits nothing relevant. At the same time, do not make a claim for involuntary denied boarding. But, do ask for reimbursement in cash for the losses.
Not that this will repeat itself for you, but for the benefit of others, if something such as this happens, call the police. This is far beyond a customer service issue (which is not a police matter) but goes to safety and security.