Plate armour making it hard for enemies to hurt you is not a bug; it's a feature.
Voluntarily lower his AC? If someone built a castle would you ask them to knock a couple of walls down to make them easier to hit?
The monsters' mistake is to keep attacking him rather than the squishies. Your mistake is trying to make monsters tougher to try to hurt the paladin, increasing chances of killing everyone else.
Keep the monsters the same, but change their behavior. Creatures change their behaviour in response to their environment. It's not like intelligent creatures don't associate plate armour with being harder to hurt.
Monsters will take the path of least resistance. Their purpose is not to kill the paladin specifically, nor to spread their attacks around the party in a 'fair' manner. They are not there to be 'fair'! They are there to...get a meal (so eat the ones that you don't need a tin-opener for), or prevent the party from invading, or reduce/eliminate the threat the party poses (kill the casters!) or..whatever.
They can simply choose not to waste attacks on the invulnerable guy. Will he feel that the fight was easy and no challenge if he survives but his mates all died?
You don't need to reduce his hit points in order to challenge him. His challenge is to do his part to help the party succeed, and if they all die then he has failed even if he is unhurt. He should be frantically trying to keep them alive. You don't need monsters tough enough to kill him, just monsters tough enough to challenge them!
As for his AC: that shield is easy enough to negate. Not by changing the rules or artificially making his choice to play a well-armoured PC meaningless, but by situations. You absolutely can say that heavy armour imposes disadvantage on swim checks! It's the DM's job to adjudicate these things in a fair way, and it's perfectly fair to say that wearing heavy armour makes swimming harder.
If he has to climb a ladder, impose penalties for using only one arm and make it almost impossible to climb a steep ladder with no hands. He then has to choose to put both sword and shield away, or choose only one of them to keep in hand. Weapons can be drawn as a free interaction, but donning/doffing a shield takes your action. Attack him as he climbs. If he only keeps his shield then he cannot attack back. If he only keeps his sword then he is -2 AC and needs an action to get it back. If he doesn't care about being attacked, wait till he gets near the top and have the baddies destroy or knock over the ladder. Armour doesn't help against falling damage and the ground isn't going to miss!
Don't arbitrarily change rules to make his choices meaningless. Instead, let him take the natural consequences of his choice to be well-armoured, both for good and for ill.