Yes you can grow basil in your climate but it should be grown in part shade so it doesn't get the hot afternoon sun. It shouldn't have wilted in a few hours inside. I have grown it in my south facing window all winter, but not in your heat. I only say this to show you it can be grown inside. Did you check to see if it needed water.

The next mistake you made is putting it outside in the sun. This plant was probably grown in a greenhouse and needed to be hardened off before going outside permanently. It had never been exposed to uv light and it has to be introduced gradually. Can you imagine what would happen to a person who had never been outside and was suddenly put outside in the sun--the same with plants.

To harden a plant off you put it outside in the shade for a couple of hours and then bring it in. you gradually increase the time it is out and gradually increase the sun. In your case and location morning and late afternoon sun.

Basil is very easy to grow from seed. It sprouts fast and is usable in a month, so get yourself some seeds. I imagine your weather has cooled a bit now so you might be able to plant it in the sun now,

The plants that stores receive from greenhouses have been watered sometimes several times a day sometimes with small amounts of fertilizer. When you got it home you were not on the plants schedule. I am aware that LV gets cold weather during the winter so I am aware that a basil plant planted outside would not live over.

What I do is purchase a basil plant or various types in the spring. Plant in the soil and use that plant for immediate gradification. I plant various basil seeds in the ground for later use. By the time the original plant either becomes mature and bitter the new plants are ready for harvest.

Personally what some call bitterness I welcome as as counter point to blander foods.