HP Spectre 13T-3000 is an excellent notebook but it comes with a internal battery that is not very user replaceable. However you can buy it on ebay by searching the part number, 734998-001. Changing HP Spectre 13T-3000 battery should take less than 10 minutes.
So long as you have the T5 Torq screwdriver to remove the bottom panel of HP Spectre 13T-3000 and a small Phillips screwdriver for the battery’s screws. I found the battery on Amazon as well for 70 dollars.
This will you replace that replace battery that is getting weak. It will also help you replace the dead battery. HP tech support could not fix it via a remote support session and says that I have to send it in but you can easily replace it at home. It is not that difficult.
Sending it to HP Support will delete everything from your computer and it will take them few weeks to get it repaired and send it back to you.