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OK, the TEE you have pictured isn't going to work. You can make this work but you're going to need to improvise and the adaptation isn't going to be pretty. Before I can tell you what to get I need to know what size the branch of your TEE is. If you don't know the nominal size I can probably figure it out if you can give me the outside diameter of the threads.
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I'm afraid that doesn't answer my question. I know what you would normally need I need to know what the BRANCH of the TEE is (apparently the connection you can't find). You've made a start in that you seem to have found a brass 3/8 compression TEE which can give you an outlet from the existing toilet supply tube but we need to adapt that 3/8" branch to the connection to the TOTO. So I need to know what size that is. Apparently you don't know so you need to measure the outside diameter of the male threaded branch on the adapter TEE that came with it.
Yes I'm talking about the gray branch. I doubt it's a unique size but anything is possible.
The 7/8" connection is unique to the threads on the bottom of a toilet fill valve. So the only way to use it would be to cut another hole in the toilet skirt that the current supply tube is penetrating . I don't think you want to do that. I wouldn't.
I need you to measure the outside diameter of the threads on the branch of this TEE. Pictures don't help unless you hold a ruler next to the threads.