I'm a hobbyist pianist who has been playing primarily rock and blues music for about 2 years, and classical before that for a few years. I have a beginner grasp of music theory (I know my major, 3 minor, and blues scales, and work on ear training and understand intervals). I thought to branch out in my playing (especially improvisation for cocktail-style piano) and get some further understanding of music theory by reading through the Jazz Theory book, however, it's pretty tough to get through for me. Given this book is nearly straight theory and has very few exercises, should I be working through other books more my level before diving into the heavy theory?
Yes. That book, while being very good, is not paced well for beginners. It throws a ton of hard-to-learn stuff at you right away. I'd recommend supplementing it with other books that perhaps have more written out examples to work with as well. I wouldn't say it's good as an only book.
The thing about learning jazz is that I could sit down with a new player and in a half hour show them exercises that would take five years to master. But that doesn't mean that's a good idea...
Have you tried Charles Austin's An Approach to Jazz Piano? It's probably the most comprehensive jazz piano/jazz theory book out there, and it starts with "rock" type harmony and moves on from there. I found "The Jazz Piano Book" assumed a lot of cultural knowledge that I didn't necessarily have at the time, when I was starting jazz piano, as well as introducing ideas in a weird order, and at the time I just needed a really thorough explanation of all the things from first principles, and ways to put them into practice.
Levine's The Jazz Piano Book is more of a textbook than a How To, so no wonder you are finding it hard work. Given that your stated aim is to get into 'improvisation for cocktail-style piano', perhaps start by obtaining jazz piano arrangements of some cocktail-style tunes; you are bound to learn something just by playing them, and the cocktail quaffers are unlikely to care whether the solo is yours, Hal Leonard's or Mulgrew Miller's (Levine is a big fan).
Next, it's a matter of experimenting with different chord voicings and slipping in a few substitute chords to put your stamp on the tune. It's probably best, when starting out, to confine your experimentation to the solo section, rather than the head, so as not to cause too many ripples in the martinis. As Laurence Payne says,'...they still want to hear the tunes...'
The next step is pretty much mandatory: Get The Real Book Software, which contains a lifetime's worth of jazz tunes and three lifetimes' worth of 'definitive' recordings. You then listen, listen, listen, transcribe, transcribe transcribe and play, play play. Then transcribe some more. You can cut to the 'cocktail' chase and skip over the more frenetic tunes; that will leave you more time for transcription. Transcription is critical because it is your link to a century's worth of innovations by jazz giants.
Sooner or later, the listening and transcribing will put you in a position to make more sense of the theory when/if you return to Levine's book.