I like the name Wyatt, but in this case I'm going to have to go with Liam Rhys. I think it flows a little better than Liam Wyatt. JMO.
I think Liam Rhys flows better. Especially with your last name.
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I would go with Wyatt... I am realizing how often my daughter has to spell out her name for everyone (it is Jessie) so names that are easily spelled will be high on my list... but it is a middle name so not nearly as often needing to be spelled out... hmm... I still like Liam Wyatt
I prefer Liam Rhys, but they are both nice.