Is "that's" for "that is" really an abbreviation? The usual early modern
abbreviation is, I suppose, a written form that is a shortening of the
word as it would have been pronounced. Your choice of words ('spoiled by
an extra syllable') implies "that's" would not have been pronounced as
"that is". So if it is an abbreviation, it is an abbreviation of a
different kind, which the type attribute on the abbr element might
indicate.

So you might have
<choice>
   <abbr type="enclitic">
      <choice>
         <abbr type="standard">yt</abbr>
         <expan>that</expan>
      </choice>'s</abbr>
   <expan>that is</expan>
</choice>

and your stylesheet would have the choice to expand type="standard"
abbreviations and not the type="enclitic" ones.

Best,
Peter