Hang in there as you are doing all the right things.
Please confirm if you have or have not determined if the 5th Div Assoc has members who served in the 11th?From your note it looks like they did.Call Carl and ask him for a list of 11th Inf Reg guys who are in his association?If they exist start calling and ask if they can help with Co F info or shed any light on your research.
It does not matter if they are Co F guys or Company A.You just need to keep speaking with survivors from the 11th as sooner or later someone is going to have the information you are looking for.
You mentioned a 5th Division year book as well?? If this is what I think it is it should have the names of other 11th Inf Reg guys and a breakdown of information regarding KIA, purple hearts awarded, medals etc ALL BY NAME AND UNIT.
If this book has this your other avenue is to search the award appendixes searching for unique CO F names and then do an internet phone search. (I did this with great success when researching my Uncle who was KIA in WWII). Then follow up with phone calls hoping that you have the right guy and that he is still alive.
The main thing is to keep finding any reference to the 11th Inf Reg and subsequently looking for the names of any men who served in CO F in any capacity. One clue will lead to another etc...