Wow that's a blast from the past, let me check if all of the information there is covered...
Yeah, looks like it is. Battle stats aren't computing correctly? You mean it's reading it as if it is a Bow/Arrow deal rather than a stand alone weapon?
I'll need to dig deeper than to look at my old notes on this matter. This goes into unexplored territory with weapon graphics involved and those are never an easy trip; but should they be figured out there is a reward to be had for doing so.
Hmm, well going into battle and setting the Poison Arrow as a Two-Handed Weapon with a Chakram appearance displayed and acted perfectly. So Two-Handed Weapons work optimally.
Or so it looked... but the game then reads the Crossbow as ?20 Arrows? and that is added onto the Poison Arrows. The Poison Arrows switch fine with the Crossbow, but the Crossbow does what I just mentioned. That will need to be looked at...
There's a lot to record here; this could take a while. I'll edit the post when I have something of value.

Here's some interesting information! It not only includes the Quiver Check, it also is the amount that will be placed into the Quiver.
$02/951A 20 9B 94 JSR $949B [$02:949B] A:0004 X:0020 Y:0000 P:envMxdiZC - Jump to Arrow Quiver Subroutine
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$02/949B B9 1B 32 LDA $321B,y[$7E:321B] A:0004 X:0020 Y:0000 P:envMxdiZC - Load Item that is to be switched.
$02/949E C9 61 CMP #$61 A:005A X:0020 Y:0000 P:envMxdizC - Is it lower than 61 (shields)?
$02/94A0 90 02 BCC $02 [$94A4] A:005A X:0020 Y:0000 P:eNvMxdizc - If so, branch past.
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$02/94A4 C9 54 CMP #$54 A:005A X:0020 Y:0000 P:eNvMxdizc - Is it lower than 54? (Arrows)
$02/94A6 90 06 BCC $06 [$94AE] A:005A X:0020 Y:0000 P:envMxdizC - If so, branch past.
$02/94A8 A9 14 LDA #$14 A:005A X:0020 Y:0000 P:envMxdizC - Load 14 into A (The 20 used for the Quiver)
$02/94AA 85 04 STA $04 [$00:0004] A:0014 X:0020 Y:0000 P:envMxdizC - Store A in 04 (Yes, Byte 04)
$02/94AC 38 SEC A:0014 X:0020 Y:0000 P:envMxdizC - Set Carry Flag.
$02/94AD 60 RTS A:0014 X:0020 Y:0000 P:envMxdizC - Return
Disabling that check (What was the deal with the 20 Quiver anyhow?) Might fix the infinite arrows bug I was just having, let's take a look...
Indeed! That fixes the infinite arrow bug! That's one hurdle jumped... as always just make sure that you have the value of 54 set to whatever your Bows start!
Now let's see what's going on with the rest of the arrows...
Now I see nothing wrong from a glance oddly enough. I've turned the Poison Arrows into a Bow with the Mute Arrows and they are acting like proper Bow and Arrows. Did you change the visuals of your new Bows to that of Bows rather than Arrows? The attack of the weapons are not being added appropriately however. It seems to only read the attack power of... frankly I'm not sure. It is 21 which is neither the power of the Arrow or the Bow. The bow is 80 and the arrow is 30. This oddity is reflected in battle and out of battle. This implies another CMP that I must have missed at the time. I'll post more as I find more.

Another miss on my part for proper decrementing of Arrow Weapons. When an Arrow outside of battle(changed into a bow) is put into another of the same Arrow (Bow) it will stack. It's likely a CMP I didn't notice because I didn't try to stack two before.
$01/C233 C9 54 CMP #$54 A:005A X:0000 Y:0033 P:envMxdIZC - Is it Arrows or higher?
$01/C235 90 06 BCC $06 [$C23D] A:005A X:0000 Y:0033 P:envMxdIzC - If not, branch to return..
$01/C237 C9 60 CMP #$60 A:005A X:0000 Y:0033 P:envMxdIzC - Is it shields or higher?
$01/C239 B0 02 BCS $02 [$C23D] A:005A X:0000 Y:0033 P:eNvMxdIzc - If not branch to return.
$01/C23B 38 SEC A:005A X:0000 Y:0033 P:eNvMxdIzc - Set Carry Flag.
$01/C23C 60 RTS A:005A X:0000 Y:0033 P:eNvMxdIzC - Return
Changing this to your desired start of Arrows should fix that behavior. It still does not fix the attack power. That is likely yet another check I am missing.

I may be onto something here...
$03/A325 BD 33 20 LDA $2033,x[$7E:2033] A:000C X:0000 Y:0002 P:envMxdiZC - Load Right Hand Weapon
$03/A328 F0 4E BEQ $4E [$A378] A:005A X:0000 Y:0002 P:envMxdizC - If it is 00 branch to near End.
$03/A32A C9 4D CMP #$4D A:005A X:0000 Y:0002 P:envMxdizC - Is it Bows or higher?
$03/A32C 90 4A BCC $4A [$A378] A:005A X:0000 Y:0002 P:envMxdizC - If not, branch to near End.
$03/A32E C9 61 CMP #$61 A:005A X:0000 Y:0002 P:envMxdizC - Is it Less than Shields?
$03/A330 B0 46 BCS $46 [$A378] A:005A X:0000 Y:0002 P:eNvMxdizc - If not, branch to near End.
$03/A332 BD 35 20 LDA $2035,x[$7E:2035] A:005A X:0000 Y:0002 P:eNvMxdizc - Load Left Hand Weapon.
$03/A335 F0 41 BEQ $41 [$A378] A:005B X:0000 Y:0002 P:envMxdizc - If 00 branch to near End.
$03/A337 D0 41 BNE $41 [$A37A] A:005B X:0000 Y:0002 P:envMxdizc - If not 00 branch to End.
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$03/A37A 60 RTS A:005B X:0000 Y:0002 P:envMxdizc - Return
This is rather interesting as it looks to check if the player has both Bows and Arrows. But why the failure? Hmm, I'll have to look through this with a proper Bow equipped to find out.
Alright, I found this for the Right Hand (Arrows in Right Hand- Bow in Left)
$03/8F41 C9 4D CMP #$4D A:005A X:0000 Y:0007 P:envMxdIzc - Is it Bows?
$03/8F43 90 12 BCC $12 [$8F57] A:005A X:0000 Y:0007 P:envMxdIzC - If not, branch past.
$03/8F45 C9 54 CMP #$54 A:005A X:0000 Y:0007 P:envMxdIzC - Is it Arrows?
$03/8F47 B0 06 BCS $06 [$8F4F] A:005A X:0000 Y:0007 P:eNvMxdIzc - If not, branch past.
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$03/8F49 A9 80 LDA #$80 A:005A X:0000 Y:0007 P:eNvMxdIzc - Load 80 into A.
$03/8F4B 85 B8 STA $B8 [$00:00B8] A:0080 X:0000 Y:0007 P:eNvMxdIzc - Store A in B8.
$03/8F4D 80 08 BRA $08 [$8F57] A:0080 X:0000 Y:0007 P:eNvMxdIzc - Branch always (Presumably for a check later on?)
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$03/8F4F C9 61 CMP #$61 A:005B X:0000 Y:0007 P:envMxdIZC - Is it Armor or Above?
$03/8F51 B0 04 BCS $04 [$8F57] A:005B X:0000 Y:0007 P:eNvMxdIzc - If not, branch past.
I can't tell you what it's exactly doing, but this is some sort of check to see if the item in the Left Hand is a proper bow. I imagine that the Right Hand Check should be close by and that should settle this matter outside of battle.
And here it is for the Left Hand!
$03/8F27 C9 4D CMP #$4D A:005A X:0000 Y:0007 P:envMxdIzc - Is it Bows or greater?
$03/8F29 90 12 BCC $12 [$8F3D] A:005A X:0000 Y:0007 P:envMxdIzC - If not, branch past.
$03/8F2B C9 54 CMP #$54 A:005A X:0000 Y:0007 P:envMxdIzC - Is it Arrows or greater?
$03/8F2D B0 06 BCS $06 [$8F35] A:005A X:0000 Y:0007 P:eNvMxdIzc - If not, branch elsewhere.
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$03/8F2F A9 80 LDA #$80 A:005A X:0000 Y:0007 P:eNvMxdIzc - Load 80 into A.
$03/8F31 85 B7 STA $B7 [$00:00B7] A:0080 X:0000 Y:0007 P:eNvMxdIzc - Store A in B7
$03/8F33 80 08 BRA $08 [$8F3D] A:0080 X:0000 Y:0007 P:eNvMxdIzc - Branch elsewhere (Likely B7 is some sort of storage for a genuine Bow check)
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And this is finished! These same values are used in battle and this solves all of the problems mentioned!
I am glad to have finished this old undertaking and feel rather embarrassed that I left so much loose in the first place.
If you need anything clarified or such please let me know.