Problem?
In this particular case, nothing. Any concern you might have about "overpowered" is mitigated by his stats being at unimpressive levels. From your previous question you indicate that the players are not leaning toward min-max, power gaming, nor optimization in a munchkinny way. He's using finesse weapons that in time may limit highest damage potential: but if they aren't in min max mode, that does not seem to matter to the group.
At higher levels, he may get a little frustrated with melee damage since there are no two-handed finesse weapons and there are fewer choices in weapons ... but a lot of that gets into what fighting styles he uses and chooses. (And you control what magical weapons, if any, become available so you can tailor that to this party/team. Our DM did just that for our Bard, who is a rapier using singer).
At higher levels he'll benefit from those Divine Smite nova shots. When paired with sneak attacks from his rogue levels, he should be able to keep his damage in a "good enough" zone if the party fights well as a team.
The other place a low Strength would be a Paladin issue is in armor selection: if he wants Chain, Splint or Plate, he needs 13 / 15 / 15 strength respectively. With the proposed Dex build and Rogue multiclass, he's unlikely to want them since they give Disadvantage on Stealth checks. He can wear Breastplate with no minimum Strength requirement -- allows up to +2 of his Dex AC bonus, and no disadvantage for stealth. At 400 GP it may be a level or two before he can afford it. A 50 GP chain shirt may be his best for the time being.
From @Ethan's comment: With leather armor, a shield, and 20 Dex (After a ASI's) he will end up with 19 AC, 1 lower than full plate+shield, but the Piradin will be good at all +Dex skills. This makes for solid function when full plate isn't a good idea -- such as waterborne adventures. His damage output will be identical to a strength based shield using paladin (barring difference in MC class features).
If this is a role and story based move, all the more reason to go with it. It sounds like fun - which is why we play. As long as the player knows where the difficulties are up front with the stats as is, the player is taking on a challenge. Good on 'em!
I'd recommend a ruling to let him proceed as desired. (And if later on they finds some gauntlets of ogre strength ... )
Only a little, if at all.
A +3 Spiked Armor that applies enhancement to Battlerager features essentially serves a similar function of a Defender that always activates (since you only get one attack). So we can use the differences to measure balance.
The first time you attack with the sword on each of your turns, you can transfer some or all of the sword's bonus to your armor class, instead of using the bonus on any attacks that turn.
Not being able to use the weapon bonus on multiple attacks is a significant downside, but you also get the benefit from the other Battlerager features.
Here is a cost-benefit analysis of the tradeoff for comparison.
- Damage per round: 3 damage per Extra Attack vs 3 damage per attack that hits you.
- I would say these are comparable if not in the favor of Defender as smart monsters will avoid attacking you in melee (most have a ranges or magic option at the level where you usually start getting magic armor).
- Armor bonus: 3 if you sacrifice your extra damage vs 3 always
- This shows the strength of the variant magic spiked armor. You have a consistent armor benefit. This is where the slight imbalance comes from, but it is probably not enough to unbalance the game substantially.
- Grapple damage: Grappling in 5e is very niche but the 3 extra damage does confer some small benefit compared to the Defender.
None of these small differences would push the weapon above or below the power level of some of the other legendary weapons (of which the Defender is included). While it is not a direct comparison for the lesser magic armors (with +1 and +2), the principle can be extrapolated to include them.
If you find that this is unbalanced since it is bonus damage for a non-attunement item, You could make the Battlerager benefits require attunement, but I doubt it will be necessary.
Best Answer
The multiclass prerequisites, in my opinion, are best treated as suggestions rather than requirements. They do not serve any balance function—they serve primarily to help prevent players from falling into the trap of taking a class that their ability scores will not support. The reciprocal nature of the requirements, in my opinion, have more to do with making things symmetric and minimizing the degree to which the order you take classes matters than it does to do with anything concerned with balance.
As such, I have never stopped a player who knew what they were doing from doing any multiclassing they liked. This has caused zero problems, and I see no way that it could. The worst case scenario, as far as I can tell, is Paladin/Warlock using the Hexblade patron, which can now be done without needing Strength 13, which is quite desirable for this combination since Strength does little for them thanks to Hex Warrior. However, Hexblade is widely considered to be an anomaly in its own right, which means it would be quite sad indeed to limit everyone else for the sake of this one case. For me, even this combination has not caused any real problems. There is certainly nothing about a Dexterity-based Barbarian/Fighter that is going to cause more problems.