Should I allow a characterb to switch two stats to make multiclassing requirements?

Just let them switch the stats and pretend it was always that way

I like the headband idea. It could have some rare stuff you have to go find in order to earn the class.

It is kind of a long path to get 8 up to a 13 and I do agree with you that just handing it to her isn't really fun. If there was a downtime system like Adventures League you could maybe let her read some books or something.

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It really depends on the table but it's also possible for you to allow the multiclass despite the stat with the aid of her friendly NPC. I'd imagine it wouldn't play to powerfully with the negative modifier but that could also lead her on a quest to fully unlock the potential of the class by finding gear to increase her int.

5e's fixed stat items should never have left playtest. I do like your idea of going to 13, but it's still a +5 boost.

Perhaps the artificer had an intelligence enhancer 9000 pro rgb lying around, waiting for a test subject. You lose wisdom, of course, but who needs that?

I'd just ignore the multiclass requirement. The requirement is there for a reason and is to prevent dips in bizarre classes to gain class features without making your character at least a little bit MAD. In this case, if you trust your player, and don't really see a reason or way in which this can be abused, just ignore the requirement. Yes, she may be a very below-average artificer, but if that's something she wants to do, and has the tutelage of her NPC friend to do so, I'd say go for it, if she wants she can eventually train and get over that handicap. Besides, the artificer isn't really that dependant on high INT stats or spell saves or attack bonuses. It'll give the Bard a little more utility and spells, and later on the mechanical construct, which (if I remember correctly, I may be completely wrong here) doesn't scale or depends at all from intelligence. If I had to guess you player's character will remain mostly a bard.

Just let them switch

I actually like using wisdom instead and rather that using intelligence to make sophisticated tech, and instead use "common sense" and jury rig up an equal yet different item.

You still use magical components, but instead of using schematics and plans and such, you just kinda slap it together based on stuff you happen to know.

Otherwise just switch them and let live. Just double check skills and saving throws and you're set

I think option 1, 2, or 4 are all viable. Just be careful with 1 that it doesn't set a precedent for everyone.

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Perhaps this stat swap item has a cost. Say, they can swap that 8 with the 14 but one ability point is lost in the process giving her that perfect multiclass 13. No machine is 100% efficient, maybe this stat swap item is a prototype and as it currently is causes a mild reduction in overall ability points.

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I like the idea of her meeting a practicing artificer who offers her the chance to take part in a case study where she gets to have her brain rattled around to improve her weaknesses.

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"Yes, please sign here, here, and here. Okay, now if this works I will surely gain tenure at XYZ Arcane Engineering Institute! After all, a machine to give the common folk the mental capacity of an arcane scholar shall surely grab the attention of the magical community!"

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*magic machine whirs up and the PC awakens many hours later*

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"Hmm it appears to have functioned mostly as intended. A little bit of your brain's capacity has been lost, but ultimately you are much more in tune with arcana than before! It has definitely become much more efficient than before, poor Alden over there..." *points to a crumpled farmer in a bed across the room who shouts " The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side!" * "Oh dear... Alden, that's a RIGHT triangle! You see, much of his mental capacity was erm... Lost in translation..."

I would like to allow it if possible since we mostly play a fun and not too serious game.

Go for it!

Option two is to allow her with the help of the NPC friend to craft a lesser version of a headband of vast intelligence as part of taking her first level as an artificer.

Or perhaps a hat? Futurama style? Please?

Option four is to say no. And she has the option of making a new character if she wants to be an artificer.

That is true, and even possibly fair. But I think letting her get a headband or perhaps a magical transformation if it makes sense could work better. I do think it should be a quest line of sorts and not easy though. If you go the monkey hat route, I would suggest it adding to int by taking away from possibly str, obviously they want to be more brain than brawn oriented. The tricky part of going magic item to achieve this, if an enemy finds out and removes it, they lose their class features.

Okay, stupid idea. Magic item that transfers "abilities" that are agreed upon to another character/npc?

Basically, I think try to make it make sense and do it. If you can't do that, tell player sorry, new character.

How about, get this death then Resurrection it could make sense and be the reason the stats swapped