disclaimer- I am from the US and have simply lived in turkey for the past couple years I am actually moving back to the states in three days. The below information is from my experiences and/or research for personal dive trips. I am not involved in the Archaeology world and the below listed organizations are simply names I have seen while researching locations and dives.
Visa: they no longer sell them at the airport anymore but you can buy a 90 day tourist visa online
https://www.evisa.gov.tr/en/. I don't know all the legalities or paperwork involved but you might want to talk to someone with more knowledge to find out if getting a filming permit could be beneficial or a complicated roadblock. Turkey has been delaying Archeological dig permits to foreigners recently as a means of leverage to get "stolen" artifacts returned. If no benefit comes from having a permit (might make it easier to get permission to dive some restricted sites) you might want to simply come as a tourist and film for personal use.
When to go: Since you are a New Jersey diver the water is warm year round in turkey (60s at the coldest and 80s at the warmest). Turkey is a cheap European vacation spot Jun-Aug so expect crowds in major tourist locations. Most places are empty Sep-May but some dive operations shut down in the winter.
Where to Stay: Hotels are cheap, Most places have numerous small hotels and several large inclusive resorts. I usually use tripadvisor.com and/or booking.com to find a place with good reviews and amenities I want (look at user pictures). If you contact dive operations they are usually associated with some hotels so they might be able to get you an even better deal.
What to see: Turkey was the center of pretty much every ancient Empire at some point so everyone has left footprint on it. It is covered in Ancient ruins and important historical sites. If you spend all your time diving you could miss out on some amazing things. Where ever you decide to go make sure to take a day or two to see local sites. Depending on your historical interests every town has an ancient castle and temple. A large portion of the locations from the bible actually are in Turkey.
Who to Dive with: I have dove most places on the southern Mediterranean coast of Turkey and on the Western Aegean Coast. For the most part I just go down to the docks and talk to dive operations (cheaper than using a booking agent) which allows me to see condition of equipment, boat, and how many people they have booked. You will want to avoid the large tourist operation and get caught up in a boat of mixed divers going to a discover scuba location. since you have specific goals you will want to contact dive operations beforehand most have websites. I you start looking at a specific company feel free to contact me and I will let you know if I have experience with them.
B
odrum Museum of Underwater Archaeology- this is where underwater archeology started (interesting documentary about it on youtube
"Nautical archaeology- beginnings (part 1)" and still the base for some important work. The museum is neat to see and It won't be to hard to get a behind the scenes tour of the work going on there and the INA facility which is in the area. The work is conducted by
"Institute of Nautical Archaeology" this is an international organization, several US colleges are members. It would probably be a good idea to contact them since they know the location of most ancient wreck sites and can get permits. Also a short walk away from the castle is one of the 7 ancient wonders "The Mausoleum". I am not a huge fan of Bodrum however since it is a spring break kind of place during the warm months and will be packed full of people and loud music and most dive operations cater more to discover scuba tourists.
Nautical Archeology Society- I believe they are UK based but are involved with a turkey based organization "
Underwater Research Society (SAD)" which has several member organizations that do underwater archaeology. NAS is running a trip in November to kaş -
"Turkish Delight" they will be using
Dragoman for the diving and trips to other local sites a company I have really liked they also produce a nice dive guide for Kas. I cannot find a link to it right now but I know Kas also has its own local archaeological society Dragoman can probably put you in contact with them.
Kaş- This is my favorite location because it is a couple hour drive from the two closest airport (Antalya, Dalaman) and was isolated until the 1970s. This led to slower tourism industry development and lack of commercial industries. The dive sites around kas where protected because of the isolation and are now protected because it is recognized they attract tourism because they are in such good condition. Kas has a diverse collection of sites, walls, caves, tunnels, shallow and deep reefs, ancient wrecks, old wrecks, new wrecks, artificial reefs, three airplanes (a purpose sunk c47, a Cessna type plane, and a ww2 Italian torpedo bomber shot down). The town is a nice quiet Mediterranean fishing village that caters to a more sophisticated tourist and is never really packed or hectic. I have dove with a number of different dive companies in Kas and found them all to be exceptional. Kas offers more support for divers than anywhere else in Turkey. Many offer nitrox and several can support tech diving and rebreathers.
I apologize for the rambling and general answers hopefully this is of some use to you. If you want to know about a specific site or company feel free to contact me and I have experience with them I will be happy to provide my opinion.
-jason