How do you handle monsters with non-magic weapon resistance?
The same way everyone else does, I expect -- the monsters take half damage from non-magic weapons. Do you think there's a relationship between specific monsters having resistance to non-magical weapons and not being hit?
Do the players find a way to get magic weapons and use other tactics to deal with them?
In general, I think most games feature players getting magical weapons. I haven't had a game yet where players were saying they needed to quest for magical weapons so that they can kill something with resistance to non-magical weapons.
Also, if it does come up that they face a monster with resistance to non-magical weapons without having any, it's because I've placed a specific challenge (in a plotted game) and I've chosen the challenge to be reflective of their abilities. In my more sandboxy games, I telegraph such things so that players can make informed decisions. If they decide to go after werewolves with normal weapons, that's on them.
Or does that break the game?
Players looking for magical weapons, or resistance to non-magical weapons? No, for either. Resistance to NMWs is something that's an early game issue and is well dealt with by the ruleset. It effectively increases hitpoints for monsters that generally have lowish hitpoints for their level anyway (note the low CR monsters that have this, they typically have the hp of monsters a few CR below them because of the buff of resistance -- to players with magical weapons, they're pushovers).
Are Displacer Beasts completely broken?!?
No!11!eleven! Displacer beasts have low ACs (13 IIRC). This means their disadvantage inflicting trait bumps their ACs up a few points. Displacer beasts would be just about as deadly as they are if they had AC 15-16.
If something has a very high AC already, though, disadvantage is startlingly good. If your opponent needs a 5 to hit you, disadvantage imposes a difference of 20% in your hit rate (from 80% success to 65% success). If your opponent needs a 10 to hit you, disadvantage is roughly a 45% drop in chance to hit (30% chance with disadvantage, 55% normally). If your opponent needs a 15 to hit you, it's a 75% reduction (from 25% normally to just over 6% with disadvantage). If your opponent needs a 20 to hit you, the drop in chance to hit is 95%. That's 95% fewer hits than without disadvantage.
Displacement isn't an autowin, by any stretch, but it's effect on high AC characters is disproportionate.
In your encounter with Giants, have them attempt to grapple the Cloaked player. Have a dragon use it's Legendary Action Wing Attack to good effect. The MM is a good resource but monsters should be more than a rote set of actions in a stat block.
A giant grappling has to get into melee, which isn't hard, depending, but is a restriction, and has to sacrifice all of his actions to make the attempt, which, sadly due to the weird grapple rules, isn't automatic. Stone giants, as the only giants with athletics proficiency, are very decent at grappling. The others, not so much. A hill giant with a +5 strength against a +1 strength cleric (assume a high enough STR to wear heavy armor effectively) yields only an average 62% chance of success. A Storm Giant with +9 STR jumps that up to 80%. That's if the Cleric isn't proficient, doesn't have a higher STR, or doesn't anticipate grappling giants and avoids melee and/or casts Freedom of Movement (which completely eliminates grappling as a successful mechanic).
EDIT: disregard the numerical analysis here -- for some insane reason I forgot most giants are actually trained in athletics. The points on melee avoidance and FoM stand, the rest I retract.
You continue to beat the dead horse that the Cloak of Displacement is fundamentally broken as a reason to deny offered methods to deal with it.
Whoa, there. I've made no such claim. My only claim is that it could be broken, depending on many factors of gameplay. The assumption that the cloak CANNOT be broken is what I've argued against.
But it's not broken to any degree that many other means of magic are broken. In a game I played over the weekend we had to fight some invisible assassins and none of us had reliable methods of detecting them. They did a lot of damage but we adapted, improvised and overcame. It's what the game is about. Of course YMMV.
And I'm not saying that you can't or shouldn't do so. I'm saying that the point at which you HAVE to for the majority of encounters to offset a magic item is a point at which you can easily have a broken game. When you're expending that much effort to counter something, the problem is large enough to address out of game instead of continuing to distort your game to accommodate it. An encounter, a series of encounters, or a recurring story element that requires adaptation or non-standard solutions isn't what I'm arguming against. That is, as you not, what the game is about. I'm arguing that a Cloak paired with a high AC character played remotely smartly can cause serious problems. And that smart player is going to anticipate your general solution set and offset his exposure. Not by chance, clerics are best positions to do this with a range of easy to achieve defenses against things like elemental damage (most save spells) or utility magic that obviates lockdowns.