Add water, tea bags, and a half cup blackberries to Instant Pot and stir to combine. Add lid and lock into place. Close vent and set to Manual/High Pressure setting for 5 minutes cook time.
While the tea is brewing, add remaining blackberries, basil, honey, and water to saucepan set over medium heat. Cook, gently pressing on the blackberries to release their juices while continually stirring the mixture to combine, for 4-5 minutes. Once honey is completely melted and the mixture starts to thicken, remove from heat and set aside.
When cook time on Instant Pot is complete, allow pressure to release naturally for 10 minutes, then open vent to release any remaining pressure.
Carefully strain hot tea mixture with a fine mesh strainer. Pour the reserved liquid into a large, heat-resistant pitcher and sweeten with the blackberry honey simple syrup, to taste. Add fresh lemon juice and stir to combine.
Serve immediately over ice with fresh blackberries and a sprig of basil for garnish. Enjoy!