It's a difference of pronunciation, and listeners would hear "that'd be me."
As several commenters note, Clint Eastwood's character is assimilating the [d] sound in [that'd] so it sounds like [thad]. Whereas formally trained speakers might say a lucid two-syllable construction like /ðætəd/ or (with some assimilation) /ðædəd/ (see Cambridge Dictionary for an example), Eastwood's character says something close to /ðæd/ (which, incidentally, is Wiktionary's transcription). Since /d/ and /t/ are voiced and voiceless variants of the same sound, the voiced sound takes over, sometimes still with a vowel, and sometimes without. Hence you hear something like [thad].
Here are a few other examples of similar [thad] pronunciation rendered in transcript as "that'd." The first two are Appalachian speakers; the third is a speaker from Indiana, which I include to show this happens in multiple dialects:
"josey wales" bounty hunter– Yorik May 03 '23 at 15:15