There is absolutely no reason why this should be a barrier to obtaining a Trade Mark as long as the designer:
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holds the correct license from the typeface font owner for the use to which you want to put the logo and
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has been provided the logo incorporating the typeface font to you in a non scalable format
You are not seeking to Trade Mark the font (the designer of the font has probably already done this) you are seeking to Trade Mark the logo incorporating the font.