Therefore, when learning a new piece, I play the right hand very slowly for 4 bars - over and over until I get it right. When I'm reasonably comfortable with that, I do the same with the left hand - over and over.
Oh dear! This is not the most efficient method.
The big problem there is "until I get it right". This is wrong.
Play and hold each note/chord until you're fingers are ready to play the the next one. Only then do you play it and release the first one. It's not a question of doing it slowly, it's a question of doing it right. As you repeat the process it takes you less time to 'get ready' between notes. It is not the notes that are difficult (unless it's a big stretch etc. - See this thread: http://www.pianoworld.com/forum/ubbthreads.php/topics/2011558/Practice%20Question.html#Post2011558
) it is the change between notes. Play at a speed that allows you to get it right every time and repetition will make you faster. It has been found that 7-10 focussed repetitions are enough for sleep to improve facility so instead of doing it over and over until you get it right, just do it slow enough to get it right 7-10 times and then let sleep do the drudge work.
My question is: How perfect do I want/need to be for those 4 bars until moving onto the next 4 bars? Should those bars be perfected at the proper tempo before moving on to the next 4 bars
You should work in phrases rather than an arbitrary number of bars but you may need to break the phrases into smaller units while you work on them. Continue working on each phrase until you can play it without stumble or hesitation then move onto the next phrase but keep reviewing previous phrases two or three times each at the start of each practice session.
I would reach a fluent tempo. Somewhere around half speed for a fast piece and three quarters for a slow piece would be quite fast for a first target so slower than that but fast enough that the rhythm is recognisable.
When you're reviewing all the phrases go back and start joining them in pairs, and so on. I usually divide a typical two page piece onto between two and four sections depending on the complexity. Only when they're memorised and close to tempo will I join them up.