It does not auto-adjust.
I agree with John Campbell of the giantitp boards on this one. When the Bone Bow says that it
functions as a composite longbow with regard to applying the user’s Strength bonus to damage done with arrows shot from it. (75)
I see no reason to assume that it only applies to parts of the rules text concerning composite (long)bows and their interactions with the wielders Str bonus.
All composite bows are made with a particular strength rating (that is, each requires a minimum Strength modifier to use with proficiency). If your Strength bonus is less than the strength rating of the composite bow, you can't effectively use it, so you take a -2 penalty on attacks with it. The default composite longbow requires a Strength modifier of +0 or higher to use with proficiency. A composite longbow can be made with a high strength rating to take advantage of an above-average Strength score; this feature allows you to add your Strength bonus to damage, up to the maximum bonus indicated for the bow. Each point of Strength bonus granted by the bow adds 100 gp to its cost.
The only clarity issue here is the pricing of the modifier, which differs between the Longbow and the Shortbow
Each point of Strength bonus granted by the bow adds 75 gp to its cost.
Considering that 100 and 75 gp are the base costs for the composite long- and shortbow respectively, it can be assumed that every point of Str modifier costs the full base price for the bone bow as well, which makes the bone bow quite expensive, especially considering the feat requirement.
Should it?
On the one hand, I'd say it shouldn't, because I can't really justify it in character/in world, and my gut says no.
On the other hand, both (most) archers and exotic weapons are well known to be rather weak in D&D 3.5, even compared to other mortals martials. Giving them a boost (which mileage varies widely depending on how often you Str rating changes) is certainly not game-breaking, especially if they need to burn a feat to use it, on top of those other restrictions.
To sum up: Buy the hilariously underpriced ring of evasion and save boosting and nondetection for a party member who will benefit more for as much as they're willing to spend on it. If you can't pool funds or have no high-dex/reflex sneaky characters who would benefit, get the much more generic periapt of wisdom
Let's look at this from an abstract perspective first:
A periapt of wisdom is worth 4000 gp.
A ring of evasion is worth 25,000 gp + a ring of nondetection (less the immunity to detect thoughts , so we'll price it as a custom item) of 3*2*2000*1.5 (because it's extra) or 18000 gp (which is amazingly overpriced compared to the ring of mind shielding + a ring of fort +2 estimated at 4000 gp * 1.5 * 1.5 (fortitude boosts don't go on the ring slot.) or 9000 gp for a grand total of 52000 gp.
Thus, if you have reason to suspect that a periapt of wisdom will be available more cheaply in the future, this will inform your choice.
However. To a cleric, wisdom is all. Wisdom is all. Evasion only works if you pass a reflex save. At level 7 with a 10 dex, you have a wonderful +2 to your reflex. (selling this ring to your thief-like character on the other hand for improved evasion should allow you to arbitrage a rather inefficient market.)
+2 to fortitude is your only "real" bonus from the ring, considering how useful the rest of the suite is to your specific capabilities. (again, if you can sell it to other characters, do so for as much as they're willing to spend up to 52,000 gp).
Looking at bonus spells per day, the tradeoff is this. How often is your character targetted by save-or-lose fortitude effects versus how much would you like a +1 dc to all spells and an extra 4th level spell per day and a bonus +1 to your will save?
The answer will vary by campaign, but it would be hard to be more persuasive by the wisdom boost, save if the cost was such that market arbitrage could provide for the same bonus, later, for less expenditure.
Best Answer
If the DM's ruled that arrows for a longbow and shortbow are different, the DM should also rule that arrows for composite longbows and composite shortbows are different from those. In fact, the house rule may be even more challenging than that because the only way to use the rules as written to reach that conclusion is by consulting Table 7–5: Weapons (PH 116-7). That table lists separately bolts under each of the heavy crossbow, the light crossbow, the hand crossbow, the heavy repeating crossbow, and the light repeating crossbow, and lists separately arrows under each of the longbow, the shortbow, the composite longbow, and the composite shortbow. Under this reading, all bows and crossbows should use different ammunition if any of them do.
The DM may even think she's doing the PCs a favor by allowing them to use at a penalty arrows or bolts designed for the wrong kind of weapon! A stricter DM would've ruled the wrong type of ammunition is flat-out impossible to use with the wrong type of weapon.
TobyY. noted in a comment that the real-world advantage of a composite bow is its self-bow-like draw-weight and draw-length but a shorter bow stave, making it a safe assumption that longbows and composite longbows can share the same arrows, while Zeiss Ikon's answer discusses how some bows will be massively less effective with the wrong arrows. Either way, though, if the DM's relying solely on the weapons table, real-world evidence and experience don't matter.
However, this is a house rule
The game makes no distinction, so far as I am aware, for example, between arrows designed for a Medium creature's longbow and arrows designed for a Medium creature's composite longbow or shortbow. (Arguments can be made both for and against bigger and littler ammunition being required for bigger and littler weapons, but I'm pretty sure that's beyond this question's scope.)
The Player's Handbook, for example, in the starting packages has the half-orc barbarian, dwarf fighter, human fighter, and human paladin carry shortbows and the elf ranger carry a longbow, yet all carry just arrows. This is consistent throughout the game. No mention's made of needing different bolts for different crossbows or different arrows for different bows except in splatbooks that detail specific weapons. (Frostburn's bone bow, for example, says that it "is designed to fire exceptionally large arrows specially made for it" (75).) Even the hand crossbow's bolts cost and weigh as much as the heavy crossbow's bolts.
The exception is the repeating crossbows, both of which use 5-bolt clips of the same cost and weight that cost and weigh as much as 10 bolts for a hand, light, or heavy crossbow. (Even then, a solid argument can be made that the extra weight and expense isn't for the bolts but for the ammunition clips in which they're sold.)
Just to confirm, the Monster Manual's half-celestial paladin (144-5) has 10 silvered and 10 cold iron arrows and a masterwork composite longbow (+4 Str bonus), and the harpy archer (150-1) has 10 cold iron arrows, 10 silvered arrows, and 5 +2 arrows for her +1 frost composite longbow (+1 Str bonus). Neither entry designates the arrows as composite longbow arrows.
Finally, the Dungeon Master's Guide on Table 7–13: Common Ranged Weapons (223) lists different kinds of crossbows and bows separately but lists ammunition in simple lump sums of 50 arrows, 50 crossbow bolts, and 50 sling bullets, undifferentiated by the weapons intended to fire them.
Sometimes game designers omit what they assume readers will perceive on their own, and, y'know, sometimes readers just don't make the same assumptions the designers do, instead readers seeing what they want to see. That's not an insult, and maybe it's even become important for some reason: the DM may have big plans that hinge on this house rule ("Only the greatest of the halfling archers can slay the King of Storm Giants with this +5 giant-king-slaying Small shortbow artifact arrow!"). I don't know. I do know, however, that in such a campaign, unless playing a swordsman or wizard were equally complicated, I wouldn't try to play an archer.