I guess the question I'd ask is "Why do you need to write down any extra rules at all?"
If the entire point you're trying to get to is that you want bards to be able to cast spells while holding an instrument... then you let them cast spells while holding an instrument. Or cast spells while singing. Or cast spells while dancing. Or orating. Or any other artistic endeavor.
Because the moment you start trying to write down "official" rules... you're going to paint yourself into every corner case out there and have to keep re-writing rule after rule and rule you make.
For instance... you create the rule that says holding a musical instrument counts as a Material focus for a bard's spells, playing the instrument counts as the Somatic component, and singing counts as the Verbal. That's fine. But now... what happens when the bard wants to pull out his sling? If his instrument was a two-handed guitar, he has to let go of the instrument, pull out his sling, use the sling, and then in the next round if he wants to cast another spell he has to spend his free action putting away his sling, NOT be able to grab his guitar since he's already used his free action, and thus is unable to cast the spell. Whereas if his instrument had been a kazoo he could have kept it in his mouth and not have had to drop it and pick it up around his weapon use... or if his weapon had been one-handed he could have kept his free hand on the guitar and thus the next round he could have sheathed his weapon and then still used the guitar to cast the spell because he never "dropped" it the round previous?
It's all these nitpicky worries the bard has to have and decide on about one-handed weapons versus two-handed weapons with one-handed musical instruments versus two-handed musical instruments (or no-handed musical instruments or no instruments at all and just sing or dance etc. etc. etc.) that for me makes actually *codifying* the rules counter-productive.
I mean... if that codification really matters to you that much in your game, I think you're going to have to accept that there are good choices for the instrument of the bard's performance, and a whole heap of bad. I know personally... if I was in a game like that, my "instrument of choice" would be Dancing and an Arcane Focus, because that way I could use a weapon and cast spells without ever having to play the "draw or sheathe with the one free action" mini-game just to be able to use my class's abilities effectively.
So best of luck however you decide to go, but just know that if handedness and the juggling of weapons and instruments is how you are going to go... you're going to effectively eliminate a whole set of weapons and instruments from your potential bard's repetoire because no player is going to want to hamstring themselves like that (even if fluff-wise they'd prefer that kind of instrument.)
But... but.. you're still missing my point.
Let me see if I can make it clear:
1) I want to make Bards actually play their instruments
2) I don't want to screw the Bard over as regards spellcasting because of #1
This means, among other things, that I definitely don't want certain music instruments to be clearly superior to others. If you can't cast spells because you're blowing on a flute or playing a guitar with both hands, that would be unacceptable - nobody would ever choose those instruments. (Or worse, players would start working around the rules; binding the flute to your face, or showing youtube videos of one handed guitar players... ;-)
What I obviously do not care about is the interaction between your instrument and other items, such as weapons. The whole point, after all, is to make a guitar into the two-handed item it is and should be.
So, if you want to use a sling on occasion, obviously you wouldn't want to use a cumbersome/twohanded instrument.
But you know what? That's okay. The bard class doesn't live or die on the ability to switch weaponry.
Some bards will ONLY cast spells. For them, choosing a flute or guitar would be an okay choice ruleswise and possibly a great choice roleplayingwise.
Some bards will want to fight early and often. These bards can simply choose to not bother with an instrument at all and instead follow the default component rules.
A few bards might - just like you - want to double up and do both. Yes, for these bards, you would want a very light and small instrument, or even choose "dancing". Which is perfectly fine - no, more than fine, because that is what common sense tells you!
(You, on the other hand, is saying this almost as if there's something wrong... What is wrong with the idea that if you want free hands and a free mouth, you obviously wouldn't want to choose a wind instrument that requires both hands AND your mouth to produce music...

)
One more thing...:
If the entire point you're trying to get to is that you want bards to be able to cast spells while holding an instrument... then you let them cast spells while holding an instrument.
No, I want bards to be able to cast spells while
playing an instrument.
It's fine if bards cast spells without their instrument, but then they aren't (=shouldn't be) using the "use instrument as spellcasting focus" option. A bard casting a spell without his instrument casts spells exactly like a wizard casting a spell without a wand or a cleric casting a spell without a holy symbol.
Best wishes for mutual comprehension,
Zapp