If you are ok with the minimum amount of stuff, such as underwear, toothpaste, shampoo, and (of course!) your phone and chargers, then you can take a 25-liter backpack and stay in guesthouses.
The Lycian Way is very good for such a hike because there are many villages and you can definitely find a hotel in every village that is along your way. To do that, you need to book a hotel in advance. If you do not book it, be ready that some of the guesthouses will be full and you will spend more time finding accommodation for the night.
I must say that not all parts of the Lycian Way are perfect for such an option because you also might take into consideration shops that are available in the villages where you can refill your food provision for daily consumption. Since you do not bring camping equipment, you have to cover your daily planned destination between the villages, otherwise, there will be no place for you to sleep. If you want to hike this backpack-free way, good logistic preparation is required.
As for the luggage, which you brought to Turkey: you may leave it in a hotel or in a storage room in. a bigger city like Antalya or Fethiye, taking with you only necessities.
Then for the hike itself, you will need:
• comfy clothes,
• shoes (!), and a
• backpack where you can put water,
• a first aid kit,
• some food which you can refill in villages
• and your phone.
Trekking poles are useful but it is possible to hike without them. Also, you can take a sit mat, thermos, sunscreen, and other things that you will use during the day.