Sorry you're having problems with tuning, it's frustrating.
When I received mine, I immediately changed to a 9 -42 string set. I adjusted the truss rod, reset the claw, and adjusted bridge for a nice low action (no buzz). I leveled the bridge setting it parallel and flush to the body surface. Tuning stability was much as you described, all over the place. One thing I noticed was that the tension on the bar was really light - a flutter tap on it gave a slow, wide effect. This was a clue but I didn't realize it at the time. (BTW, if you don't get the neck relief and string height in "harmony", the high E/B strings will choke out at the ~16th fret on a full step bend.)
I finally contacted EBMM customer service and they gave me a few things to try. LIGHTLY use 600-800 grit sand paper to clean up the nuts slots, 2 springs in V pattern, and lube the pivot posts/friction points. Tried everything they suggested with no real help. I finally decided to send it back to them under warranty.
They had it for about a week. They initially replaced the bridge, pivot posts, saddles, and selected tuners - still no help. They finally replaced the entire neck and that pretty much fixed it.
The rest of what write is my opinion and not anything they specifically told me. When I got the guitar back, the first thing I noticed was that the saddles on the bridge were raised significantly higher than before, while the bridge still floated level and flush with the body surface. The string action was still low with no buzz, too. The new neck looks thicker at the heel raising fingerboard "plane" and requiring the saddles be raised. Raising the saddles on the bridge plate increases the distance of the strings to the pivot post plane and requires more spring tension to balance at pitch. They also added the 3rd spring back to balance. The trem bar is much stiffer now, the tone is fuller, and most importantly the tuning is solid for a floating system. It's not as good as a double locking system, but it's good.
I tried raising the saddles before sending it back. But, that required me to lower the bridge plate deeper in the pocket to maintain string action. At that point, it lost a good amount travel and hit the pocket floor pretty quickly when using the bar.
After a week of heavy playing, it was now time for a restring. I replaced the strings with a regular EB 9-42 set (pink pack). Stability wasn't as good as before. I changed the strings again and used EB RPS 9-42 set - tuning stability was back. I've also noticed the more I play it, the better it gets
Hope this helps. Message me if you want more detail.