@deb23

These are all excellent questions! I have ILC, diagnosed de novo Stage 4 with mets to bones.
1. " is the radiologist experienced or trained on what unusual things and in what unusual locations to look for because this is ILC?"
They probably are. I always feel bad that patients do not have a way to communicate with the radiologist reading the scans. I once challenged a scan with my oncologist. She asked for a re-read, and asked around a second person to read it as well. One thing you can do is, ask your oncologist to put a note when requesting scans that you have ILC. Perhaps it will give them a heads up.

2. "PET scans.": I think there are different schools on this. My onco never ordered a PET for me. I asked about it multiple times but she said she believes "CT CAP + bone scans are the best way to follow it" in my case. Now bless your heart, I already have bone mets, so it may be a different scenario for me. There are some PET scans that show the Estrogen+ cancer in the body, you might want to ask about them (18F FES PET). Some dr's caution against the glucose your body intakes for PETs. Again, there are different schools...

3. CT: CT will, in all likelihood, show if there's something in your organs (kidney, liver, etc.)

4. " I ask and ask questions and am probably becoming a pain in my onc's neck": It's good, you will keep them on their toes 🙂 In my dr's notes it says "the patient has anxiety over her diagnosis". You bet I do!

I hope and pray it never comes back to you. Ask questions, take some actions, sure, but please do not over concern yourself with thoughts of stage 4. It looks like you are under good care. All research agrees on good effects of regular exercise, eating healthy, sleeping well. Try to adopt these and hope for the best.