I'm building a very gimmicky multiclass crit-confirmation paladin, and as a part of my build I have a feature that lets me reroll a failed attack roll as a reaction. The idea is that I want to fail at least one attack roll per turn so that I can roll again and potentially get a crit; but the thing is, I'm not sure if I can intentionally not add proficiency bonus to the attack roll of a weapon I'm proficient in. If I'm on the last attack on my turn and haven't missed yet, can I choose to make a sloppy attack as though I didn't know how to use a weapon to get that extra chance at crit confirmation?
Multi-class into (or hang around with) a Divination Wizard
Starting at 2nd level when you choose this school, glimpses of the future begin to press in on your awareness. When you finish a long rest, roll two d20s and record the numbers rolled. You can replace any attack roll, saving throw, or ability check made by you or a creature that you can see with one of these foretelling rolls.
Use your low rolls to make sure you miss. Admittedly, this only works a limited number of times, but it gives you the best chance to assure a miss.
The proficiency bonus just gets added but...
As worded , the proficiency bonus simply gets added to your attack rolls:
Proficiency Bonus: You add your Proficiency Bonus to your Attack roll when you Attack using a weapon with which you have proficiency, as well as when you Attack with a spell.
But you can simply switch to a weapon with which you are not proficient if you want to perform a less accurate attack.
There is nothing in the rules that allows for this.
The rules for making an attack state:
You add your proficiency bonus to your attack roll when you attack using a weapon with which you have proficiency, as well as when you attack with a spell.
The language here does not make this optional, and there are no rules elsewhere that make this optional.
You might be able to give yourself disadvantage by closing your eyes.
Disadvantage can easily be gained:
When you attack a target that you can't see, you have disadvantage on the attack roll.
Announce, “I close my eyes before swinging my sword”, and you get disadvantage on the attack roll, but your mileage may vary depending on how the feature is worded and how annoyed your DM is with this tactic.
Also disadvantage means you have almost no chance at a crit on the initial roll (1/400).
Attack with an improvised weapon
As a paladin, you would be hard-pressed to find a weapon that you are not proficient with. However, you can attack with an object that is not a weapon at all (say, a rock you found on the floor or even your shield), turning it into an improvised weapon. Unless you have the Tavern Brawler feat, you are not proficient with improvised weapons.
Throw your weapon
In case you do not have anything at hand that is not a weapon (or no free hand), there is still another option available to you: You can throw your (melee) weapon. Just make sure that it does not have the thrown property by accident and it will again be treated as an improvised weapon for this attack.
Just remember that you then don't have a weapon anymore, until you retrieve it or draw another one.
As other answers have stated, there is no way to not add the proficiency bonus while attacking with a weapon you are proficient with, but here are two other options to increase your chance to miss with an attack.
Option 1: Attack with a poor score
Depending on other factors of your build, make either Strength high and Dex low, or Strength low and Dex high, and choose a finesse weapon (which can use either Strength or Dex for attacks). Then on your normal I-want-to-hit attacks, attack with the high score and its associated bonus, and on your final I-want-to-miss attack, attack with the low score and its associated penalty.
Option 2: Human shield
Hire a squire to be interposed between you and your enemy, thus granting your enemy either half (+2 to AC) or three-quarters (+5 to AC) cover, depending on the size of the squire. This has the advantage of working in conjunction with option 1, but the disadvantage of likely burning through squires at an alarmingly high rate, so be sure you have a good supply.
A target with half cover has a +2 bonus to AC . . . if an obstacle blocks at least half of its body. The obstacle might be a low wall, a large piece of furniture, a narrow tree trunk, or a creature, whether that creature is an enemy or a friend.
A target with three-quarters cover has a +5 bonus to AC . . . if about three quarters of it is covered by an obstacle. The obstacle might be a portcullis, an arrow slit, or a thick tree trunk.