Should I start using my Mars Hydro TS300?

Sort answer yes! The LEDs will help to hardness off and acclimate them to be higher intensity of the sun outside, but you're probably good to start putting them out in a dappled sun / semi shade area as of now anyway (May24) :) If it's +10C a night there, you're fine to leave them out overnight, if not bring them into the shed /garage. After another week or so outside they should be just fine in the ground or pots. Happy growing!