I will be flying to the Middle East in a few weeks. Turkish Airlines has amended their policy, and they’re not letting anyone take any carry-on. They will move your carry-on to the cargo alongside your checked luggage for no charge. But since I never let a camera and lenses out of my sight, and I have to in this instance, I was wondering what the best way of packing them in my suitcases would be. Note that I have gone back-and-forth with them and they are not budging on this current policy, nor do I have access to a hard case such as Pelican, so please don’t suggest that. Thank you.
There's no reasonably safe way to place high-end camera gear in checked baggage without using a hardened, lockable container such as a Pelican or similar case. If the handling doesn't damage your gear, the handlers will almost certainly make off with it.
You could ship them ahead of time via another, more secure method with a small shipment specialist, but you're going to need to use the same kind of protective container to do that, too. FedEx, UPS, DHL, etc. will probably provide some sort of protective container for an additional charge if you ship it via their service.
If you don't have any possibility of using a hardened case, then your only real option is to find another means of getting from wherever you are starting from to wherever you are going.
In the hold of a plane, your equipment will be subjected to some pretty harsh conditions that can permanently damage it, so pretty extreme precautions are usually required.
I know you asked not to suggest them, but Pelican cases of a reasonable travel size would not be a huge investment. I paid less than US$200 for my carry-on sized travel case. A small price to protect the US$8000 of equipment I've usually got packed inside it. I've had very good luck packing my gear in Pelican cases for air travel, including trips to the middle east. You can either use a small enough case that it fits in your luggage or send the case on its own. (Aside from a few scratches on the outside of the case, I've never had anything inside the case damaged.) Hard cases are the only way I can actually recommend for checked luggage.
That being said... If that is truly not an option: at the very least, you'll want to pack them well padded in something that you can seal against moisture and dust.
Either way I recommend you pack them with silica gel packets (20g minimum per cubic foot with an extra vacuum sealed set to prep for the trip back) a day or so before your trip to dry out the inside of the container. This will prevent the temperature and pressure changes from causing condensation during the flight. Also place them near rigid features in your bag to protect them from crushing as bags are stacked on top of it.
It never hurts to request that you check and retrieve your bag plane side, some airlines will allow this, others won't. Some airlines (for a small fee) will allow you to to do secure pick up from the airline office where they will require ID for pickup. If they require you to check at the desk and pick up at baggage claim, make sure you high tail it to baggage claim as soon as you get off the plane so you can catch your bag the moment it comes onto the conveyor.
I've carried film kit in checked luggage before by wrapping them in as many layers of clothing as I can in the effort to protect them from shock.
In my case, the camera was late 70s/early 80s mechanical kit, and had no electronics to speak of and was absolutely fine. I have no idea how a more complex DSLR will fair though.
If a hard case or leasing isn't possible, are you able to source another camera/lens that you can take with you in place of your main setup? If you can buy something second hand that you aren't so worried about risking, it might prove to be the cheapest option.... that's if you don't need anything high end while you're out there.