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 ... )

Brown Mold isn't immune to everything

It's only immune to Fire damage, and is particularly vulnerable to Cold damage, and that's it. Which means any kind of damage other than the two listed above would still damage the mold- and eventually destroy it.

• Acid? Sure, but don't expect it to clear a 10x10 patch of it, so you'd need a lot of acid.
• Bludgeoning? Slashing? Maybe. How exactly does the player want to achieve it? (Hint: A Ten-foot pole is required)
• Piercing? Ugh... No.

As always, use your best judgement on how effective a certain type of damage is. Be open to proposed solutions and reward player ingenuity, if it's fun for the table, you're doing it right.

Additional guidance on Object's Hit Points and AC can be found in DMG 246. I would personally rule that a patch of Brown Mold is a Large Resilient Object with similar AC to soft objects. Based on the tables, that puts it at AC 11, HP 27.