A good way to get calibrated about when any bonsai needs water is to not water until it loses turgidity. As soon as you see the foliage sag and get 'weepy' make note of the look and feel of the substrate and your monitor (ribbon, chopstick, etc.). Then water immediately – turgidity will return within an hour or two.
I keep my Japanese pepper tree in an inorganic substrate called Turface MVP. I use it simply because it is inexpensive and readily available where I live. Volcanic 'soils', pumice, scoria, and lava rock in particle sizes of about 2mm may be the best substrate, but just about any porous inorganic will do. The advantage of these is that they retain very little water between the grains, so a lot of air gets to the roots and it is nearly impossible to overwater - you will have little reason for your current worries.
I live near the water between Seattle, WA and Vancouver, Canada. It is generally overcast and raining except in the summer. My tree sits outside on a shelf (on the ground when there is freeze danger) exposed to the elements. Sometimes it is being 'watered' for days on end by the rain with no problems. Organic substrates (potting soil and the like), retain quite a lot of water and the danger is that the roots don't get enough oxygen (i.e., that they literally drown).
Regardless of which kind of substrate you have, there will always be a zone or layer at the bottom of the pot that is saturated with water. The smaller the grains the higher/deeper this saturation zone will be. It is there because of capillary action – water is held against gravity by the wetting attraction of water to the particle surfaces. So after watering you can tilt the pot on its side (not so far that the medium starts to fall out). This will reduce the saturation zone. I often do this for the first few weeks after repotting, returning the pot to its normal stance once the water has stopped dripping from the pot. And, when it is expected to be raining heavily for an extended time, I prop the pot so it sits on a slant for the time.
I’ve had my pepper tree in several different depth pots. Last year it was in the shallowest, one about the same depth as yours and it was not as vigorous as it had been in deeper ones. So, I moved it to a pot about 5 cm deep when I repotted it a month or so ago.