The standard pronunciation for bitter is [ˈbɪtɐ], i.e. the phonemes /ər/ are represented by the vowel [ɐ]; for bitte, it is [ˈbɪtə], only differing in the final vowel, but clearly distinguishable for German native speakers.
The pronunciation [ˈbɪtər], as heard in your linked example, is very unusual, probably only used when over-enunciating. It is recommended in Theodor Siebs’s Bühnenaussprache (pronunciation for theater actors), which was described as “ridiculous” already a hundred years ago.
On the other hand, [ˈbɪtr] without an ə (or with only a very weak one) does occur in several regional variants of German, e.g. in Switzerland.