The Quick Solution
If your players want a helmet to look cool and also have some function, the simplest solution is to say that the 'function' the helmet provides is already part of their armour. For example, you rule that Scale Mail comes with a helmet, you only get the AC 14 base when you are wearing the full kit. This also avoids a potential future problem, if there is a player who wants to wear a helmet purely for aesthetic purposes, you aren't forcing penalties/bonuses on them that may interfere with their build and play style.
The Homebrew Solution
Shields
If you want to create a new item type, balanced with the rest of the game, I'll take a quick look at shields first, which function in a similar way.
A shield gives +2 AC, at the cost of being unable to dual-wield, or use a two handed weapon.The highest damage dice on a one handed weapon is 1d8 for an average of 4.5 damage. For a two handed weapon the highest average damage roll is 7, from 2d6. It is restricted by proficiency, a character without shield proficiency cannot effectively fight while wearing a shield. It is worth noting which Unarmored Defence class abilities are restricted by shields. A barbarian can wield a shield and still gain the benefits of Unarmored Defence, but a monk cannot.
Shield Summary:
- +2 AC
- -2.5 average damage, per attack.
- Barbarians can wield a shield and benefit from Unarmored Defence, Monks cannot.
- Requires Proficiency.
Proficiency
Starting with proficiency, I suggest splitting your proposed +1 and +2 helms into light and heavy. Light helmets require proficiency with light armour. Heavy helmets require proficiency with heavy armour.
Exhaustion:
As it stands, your proposed rules for exhaustion are weak, and finicky to track. To avoid it a player has to simply remove the helmet for a few minutes every three hours. A two minute break a character takes when there is no immediate threat is not normally something tracked at the table. As such it is unlikely to have any impact on the game, unless you are springing encounters on them whenever they take their helmet off, which will feel cheap. If your table likes keeping track this level of detail, go for it, but I don't consider it to balance +1 AC.
Perception and Investigation Penalties:
As others have pointed out, penalties applied to ability checks in 5e normally take the form of disadvantage, roughly the equivalent of -5.
I have never needed to do an investigation check in combat. Taking the helmet off for investigations, and putting it back on for combat does not balance the +1 AC. Perception is similar, except it has some chance of being used in combat, if you need to perceive a hidden enemy. Both of these checks are more likely to be used just before combat breaks out, which means the donning/doffing rules should be considered, and their impact on the action economy.
If the donning/doffing is the same as a shield, one action, this prevents the helmet from being an off/on switch for exploring and combat. If attacked while investigating, the player now has to decide to fight without the helmet, or use their action to put it on. But, even taking an action to don/doff it, the amount of in-game moments that will make this an interesting choice are very few, DM dependant.
Flanking
This penalty isn't well defined in your question, but I think it has the most potential to make helmets interesting items to serve a unique niche in character choices.
These is the variant flanking rules:
Flanking on Squares. When a creature and at least one of its allies are adjacent to an enemy and on opposite sides or corners of the enemy's space, they flank that enemy, and each of them has advantage on melee attack rolls against that enemy.
When in doubt about whether two creatures flank an enemy on a grid, trace an imaginary line between the centers of the creatures' spaces. If the line passes through opposite sides or corners of the enemy's space, the enemy is flanked.
If you are already using these rules, "Very susceptible to flanking manoeuvres in combat" could mean removing the restriction of being in opposite sides or corners to the character wearing the helmet. This is effectively giving all enemies pack tactics against that one character, getting advantage on attacks if another ally is in melee range. If you are not yet using these rules you could implement them with the addition that a character can only be flanked if wearing a helm.
In summary, a helmet grants +2 AC, with advantage on attacks against the wearer if surrounded. This gives helmets a unique niche. Characters have better survivability against ranged attacks when not in melee, against a single melee character, but worse against a group of melee characters. Characters who stay out of melee will also benefit the increased AC, if they are willing to gamble that they won't be flanked.
If advantage is the equivalent of +5, and bonus to enemy attacks reduce your effective AC for that attack, the helmet offers +2 AC in the right circumstance, but -3 in a bad one (2 - 5 = -3). This is a simplification, advantage isn't a true -5 and it also means enemies have nearly double the chance of scoring a crit against you, from .050 to .098 chance. Here are more details on the effect of advantage.
I'm going to denote the situational bonus/penalty as +X/-Y for the rest of this post, but don't forget the crit chance as well, especially since it doesn't matter how high your AC if one gets scored against you.
Is this final version balanced? +2/-3 seems kinda of harsh, especially since most combat encounters are created in such a way that melee characters are often surrounded by more than one enemy. If not used tactically the helmet will be a greater hinderance than benefit to its wearer.
tl;dr Give me the New Rules
Vulnerable to Flanking:
When a creature and is surrounded by at least two enemies, all melee enemies have advantage on attacks against it.
Light Helmet
Light Armour
10gp
+1 to AC.
One action to Don/Doff
The wearer is subject to the Vulnerable to Flanking Rules if the Variant Flanking rules are in use. If the Variant Flanking rules are not used in regular play, the wearer can be targeted by those rules.
Heavy Helmet
Heavy Armour
300gp
+2 to AC.
Str requirement 15
Disadvantage on Perception
One action to Don/Doff
The wearer is subject to the Vulnerable to Flanking Rules if the Variant Flanking rules are in use. If the Variant Flanking rules are not used in regular play, the wearer can be targeted by those rules.
Reasoning:
Light helmets provide +1/-4. This is a terrible choice for anyone who is regularly in the middle of the melee, but for characters who tend to stay on the edge of the fight, it may be a tempting gamble. The Heavy Helmet is at +2/-3 making it more attractive to melee fighters in the middle of it, especially if facing down a single enemy, or if the party has good battlefield control. I modelled it against the Plate heavy armour, which is where the high cost, Strength requirement come from, with disadvantage on Stealth switched out for Perception, per your original idea. Finally as types of armour, proficiency requirements are added, and monks and barbarians cannot use a helmet and benefit from their unarmored defence. The question was whether to make the same exception as shields for barbarians, but if the penalty for wearing a helm is attacks made against you with advantage, barbarians would get to use reckless attack at no extra cost when surrounded.
Final Things To Consider:
+1/-4 and +2/-3 Seem like a stingy tradeoff to give the characters. Playtesting would be required, but bumping the light and heavy helm up to +2 and +3 might be better balanced. This makes the tradeoff +2/-3 and +3/-2. +3 AC in the right circumstance is a a pretty big buff though.
This is also makes the effective AC of a character pretty swingy. A helmeted character might find themself untouchable one moment, when engaging on their terms, and a punching bag the next when the get ganged up on. These rules place a lot more emphasis on battlefield control, and letting helmeted characters choose when and how to engage with enemies. If your players aren't used to playing tactically, this will make the game more difficult. Sometimes the best tactical choice will be to not wear the helmet.