Start I2C communication at an external event
You could set a 'volatile' flag in the interrupt routine. In the loop() you can check if the flag is set. That is how the i2dev uses the interrupt for the MPU-6050. See the MPU6050_DMP6.ino example.

First thing to do is set the I2C clock to 400 kHz. Keep the wires short and pullup resistors low.

Wire.begin();
Wire.setClock(400000L);   // 400 kHz I2C clock speed


You can let the MultiSpeed I2C Scanner run for a while to see if it is stable.

The ADS1115 has also I2C High-Speed Mode up to 3.4MHz. The Arduino can not do that. The Arduino has a fast SPI mode, but the ADS1115 has no SPI bus.

Using an interrupt adds extra overhead. A fixed software timer with millis might be just as fast.

Do you want the maximum of 860 Samples per second? The Arduino Uno or Mega or Leonardo are too slow for that. Do you want to continuously read data or just a burst and store it in ram ?