Well, visually the period is just plain ugly. The point of a period is to distinguish one sentence or idea from another. There's no point in having it if the phrase or sentence stands alone, as in this case. The ">" is, in its own way, a kind of punctuation itself, and generally, it's better not to mix punctuation.
In this case, the period is distracting, so will distract the user and take attention away from the purpose of the words—which is to get the user to click the button!
If the "boss" insists on something like this, a better option might be an ellipsis (the three trailing dots). E.g.
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In this case, the arrow would be out of place, though.
Note that the ellipsis is not just three periods, but is a character in its own right. Separate it from the preceding word with a space.
You can get this symbol by clicking Option + the semicolon key (Mac) or Ctrl + Alt + the period key (PC).
You can encode it with