Other Parts Discussed in Thread: TIDA-00709, UCC24636
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I suggest that you look at the EFD20 ferrite core shape. It looks to me like it should easily handle the ~21W power level of your application, and is less than 12mm in height in both SMD and TH bobbin options. The EFD15 core size might also work, but it may require an overly-large gap which may lead to excessive copper-loss and/or create EMI issues.
I found it in the latest Ferroxcube catalog, although other manufacturers also carry such a core.