Step 1: Charge the battery.
Step 2: Insert it in the camera. Does the camera say fully charged? If not, it's likely the battery.
Step 3: If it says fully charged, how long does it take for the battery to draw down? Remove the battery wait a a number of hours, reinsert it, has the battery lost power? If so it's the battery. If the battery has not lost power then the camera has a current draw.
Most likely it is the battery. You could gamble on getting a new battery of a different brand if a Canon battery is not available. If it's the camera, because of its age (roughly 2011), Canon isn't likely repair it. If you could find someone to repair it, you'd likely spend more than the camera is worth.
I'd gamble on a new battery of another brand if a Canon battery is not available. If that's not the solution, sell the T3 body on Ebay for parts--disclosing that it drains the battery fast--and consider an R100 as a replacement camera for the very affordable price.
I have a Pentax Program Plus that I got for $10 because it's a battery eater. I just remove the batteries after a shoot. Someone may want the T3 even as a battery eater.
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