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I like the idea of trying the bag of vinegar before trying to take it apart. I'm doubtful because there doesn't appear to be any build up of crud the I look into he opening, but it's worth a try.
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Ya never know. Once in awhile you have enough DIY Karma Points and you get Lucky. And it could be that a chunk of Rust or Hard Water build up has broken loose from someplace else and gotten stuck in the Tub Diverter.
It could be that a Turkey Baster Flush Out and a White Vinegar Bag does the trick. If not, Here's a vid showing both types of Tub Faucets being Removed and Replaced.