Could TDWJL have participated in the last supper but simply not have recorded it?
I believe this is unlikely. In the fourth gospel, the author is very detailed about events and dialogues throughout the book. The account of that evening in the room spans from chapter 13 to chapter 17. He obviously had no hesitation in providing details of that evening.
The dialogue between Jesus and Nicodemus is the entirety of chapter 3. Much of chapter 4 is Jesus and the Samaritan woman. Chapter 6 is the feeding of the 5000 and the preaching related to being the bread of life. Chapter 9 is healing the blind man. Chapter 11 is the resurrection of Lazarus. TDWJL was keen on providing details throughout the book.
A response to this is that TDWJL wanted to just provide details that are not covered in the synoptic gospels. He didn’t want to rehash things that are already mentioned in the other gospels.
However, this is not likely either since he does mention things that are also in the synpotics. All four gospels record Peter’s denial of Jesus, the garden of Gethsemane, Jesus arrest, and the trial before Pilate.
It is also unlikely TDWJL used the synoptic gospels as material for his own book. It seems he wrote his gospel to be an independent account. He did not decide what to write based on what was in the synoptic gospels.
Could there be reasons why he might want to leave out the supper if he had attended it?
One theory I’ve read is the sacrament is like a secret ritual and he didn’t want to disclose what actually happens during the Eucharist. However, the cat was already out of the bag. The fourth gospel is generally accepted as being written after the synoptic gospels. He wasn’t hiding anything by not writing about it.
The fourth gospel also talks about the sacrament of baptism. So why hide the sacrament of the elements, but not hide the sacrament of baptism?
Jhn 1:26 John answered them, saying, I baptize with water: but there standeth one among you, whom ye know not;
So, all this leads to indicate he did not record the supper because he did not attend it.