I have a feedback for UI. It's good to see game is leaning towards strategy-management but i feel you dont want to overwhelm player with details, main part of the game is RPG; it is not a paradox game. So, why not make UI more like king managing, signing, sealing documents?
Percentages, info bars makes things appear in 3rd perspective(like controlling a country, in paradox games), and influence number display makes it gamey. It can still work with numbers, (like hitting with a sword works in numbers), but in sword control player feels the hit rather than calculating numbers. Right now it looks like player counts ifluence rather than feeling it.
May be we can see Kingdom Laws in a display like this and do something on it. Currently Management UI looks like game's informative UI; which implies to player he has no control over kingdom, just an observer. For example, in Warband we talked with our NPC's about kingdom stuff; which is more RPGish and immersive than current system. I am sure it can be improved.
To make influence look less gamey, i would suggest to design influence as a variable not accumulating/spendable; but rather a value flows like velocity. Instead of counting distance(current influence design), player can just can adjust to principles to increase/decrease his velocity(suggested influence design). Like hitting a sword.
For example in suggested design, you as a king have high influence; and as long as you have high influence you can reject laws in your favor. But changing this laws decrease your high influence for a period, a similar trade off but not same, in a special period where you have high influence(like war period) you can take important decisions which requires high influence. In the current system you cant implement something like this, if you do it(like gaining ifluence in war time) you can use this influence in peace time; but this is not how influence works in real life; not a good reflection.
It's like making influence income main parameter basically, and it can be designed as complex as much like designing sword physics, player only feels how to spend it and learns it. Only learning principles are enough which will be displayed, but player can master the topic with experience. Player will be aware of principles increasing, or decreasing his influence, like currently, but it is smoother and less precise of course. Like real life! If player has low influence, he cant control it instantly or use stacked influence for safe use; he has to face risks and consequences.
And if you think about it, you can make it elaborated without smothering player into details. Think of how a car moves, accelaretes in daily life, describing the physics of it is pretty complex but people intuitively understand principles of it, and it is enough. It's easier to accelarate a motorcyle compared to a truck. Similarly, a small lord winning a decisive battle can 'obtain' a important influence value compared to a already strong king; the math behind it not necessarily requires explanation. Or a car's speed will decay if there is no more power applied, similarly if you lost your title your influence will decay etc. etc.