What is the proper superlative of "bitter"?

Both forms are well represented, with bitterest being the more common of the two as this ngram illustrates:

Superlatives of two-syllable words can be formed either by inflection (adding -est) or by using the most. Some are more common than others, depends on the syllable structure.

These two are not mutually exclusive options, and both can co-exist happily, as we see in a most bitter conclusion side by side with his bitterest enemy. Both are correct, and the choice depends on the writer or speaker.

In terms of use (where the most used term is the most "proper" [properest?] or "correct" term), 'bitterest' appears 302 times in the NOW corpus (News On the Web), while 'most bitter' appears 441 times.

However, of the 302 appearances of 'bitterest', 134 are in 2016, while of the 441 appearances of 'most bitter' only 115 are in 2016. The difference may (or may not) reflect a recent trend favoring 'bitterest'.

If you're defining 'proper' and 'correct' in terms of rules, it comes down to what sounds better to you or your audience.

About the NOW corpus, the compilers say this:

The NOW corpus (News on the Web) contains about three billion words of data from web-based newspapers and magazines from 2010 to the present time. More importantly, the corpus grows by about 4-5 million words of data each day (from about 10,000 new articles), or about 130 million words each month.

With this corpus, you can see what is happening with the language this week -- not just 10 or 20 years ago. For example, see the frequency of words since 2010, as well as new words and phrases from the last few years.

(op. cit.)