I agree with veneering the bottom, but largely because of appearance. The border will help keep things flat on something like this and if the thick veneer is applied with cauls and weight, it's more like laminating two boards, rather than veneering in the traditional sense but putting a layer on the bottom will assure things stay flatter easier. This should have been mentioned earlier...my bad for missing it, but I'm glad that John brought it up.
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The most expensive tool is the one you buy "cheaply" and often...