Another angle. Depending on how much $300 means to you vs what your reputation means to you… I would refund her as soon as she has given her account of the scenario to UPS.
I always re-ship out lost packages asap to my customers straight away and I never wait for the mailing company to finish their investigation. I care about my reputation more than a few hundred bucks. Sometimes it really sucks to eat some cost (has happened less than a handful of times in the past 3 years) but those customers always are loyal after, and will give positive reviews and feedback.
It takes a lifetime to build a reputation and 1 second to destroy it. Something my grandpa used to say.
Yes you need to decide for yourself the character of the customer and if it is worth it. Bad luck it sold out otherwise you could have sent a replacement. But I always try and de-escalate angry customers as soon as possible. Have you tried to phone her? Talking on the phone can sometimes fix everything.