Once we get past the emotions, the fact is that OP presented the gate + crew with doc's instructions at the last second which required the device onboard for use "if necessary". Without an alternate power source such as battery or a reliable AC source (which US would likely have provided if given several days notice), the device would have been useless "if necessary".

Had the child needed treatment, the sole option would have been an emergency medical diversion. That is sometimes dangerous and always time-consuming and extraordinarily expensive. It is not always successful.

Showing up at the gate with a note was a huge mistake. This issue needed to be dealt with days in advance. It may seem that the agents were rude, but they do have 150 other passengers to deal with and some of those passengers have problems which can be solved.

Going by the book, US owes OP $0 in compensation. It is possible that as a customer service gesture, US might offer OP a credit for future travel if OP sends in a short & concise webform request which: 1. accepts responsibility for not dealing with this in advance, 2. leaves out criticism of staff who were following protocol, 3. and sticks to the basics. It should be 3-4 short sentences at absolute most.

I too would appeal the insurance decision, but the fact is that OP's child was not too sick to travel, but rather required equiptment which could not be accommodated because the issue was not dealt with in advance.

If this sounds harsh, consider the alternative. The carrier boards the family, an incident occurs in bad weather at altitude and the child does not survive a medical diversion. Not only is there personal loss and devastation, but in the society in which we live where the first words out of peoples mouths are "sue " (just skim FT for this) and ask yourself what you would do if you were the US medical service's flight surgeon.