This is a challenge. Japan is the most challenging of all the major factions to play for a number of reasons, but there are a few things you can do to enhance loyalty.
1. The most difficult is to win a war with a large enough war score to annex a region. There are several avenues to pursue to achieve this; however, most involve a war with China which may not be so much fun until your reach the reforms. But victories in battle really win a population over.
2. A simple thing to do is to station a regular infantry unit in each of Japan's home regions. A regular infantry unit will tend to raise loyalty each turn it is stationed in the same area. Militia and cavalry do not help. While MP units should assist your infantry units, they will not do it on their own, and yes, use all the governmental options you have to reduce militancy.
You indicate garrisons do not work for you, but regular infantry should do it.
3. If nothing else works, station a major army in each of your Japanese home islands objective cities and wait for the revolt. Then crush it.
My thoughts. Not sure your solution.