How to fix (2013 update)
I've tried the bugged saved game file I initially posted in my question with the current latest versions of the official patch, Unofficial Skyrim Patch (USKP) and Skyrim Script Extender (SKSE), and the saved game is now Aspect of Peryite-free.
I've configured SKSE to use its ClearInvalidRegistrations console command on every saved game load, which removes invalid scripts left running by uninstalled mods. This feature was introduced in v1.6.7 of SKSE:
add console command ClearInvalidRegistrations to remove invalid
OnUpdate() registrations
This prevents orphaned OnUpdate() events and
the resulting bloated/broken saves when removing certain mods. When
applied to an already bloated save, it will stop growing further and
instead shrink over time as the game processes all queued events. This
may take hours depending on the amount of bloat.
To execute automatically after each reload, add this to \Data\SKSE\skse.ini:
[General]
ClearInvalidRegistrations=1
According* to a developer of the Unofficial Skyrim Patch (USKP), the underlying cause is because DA13 ('The Only Cure' quest) is never stopped.
A possible fix* for this:
In the meantime, load an affected save, then type stopquest DA13 into the console. That ought to put an end to that. Plus it’ll free up some unnecessarily leftover alias memory.
BTW, don’t use the console command on a save not bothered by it. Even though it may appear benign, console commands can have negative consequences you don’t know about.
If this doesn't work or if you experience issues like: NPCs not responding properly to conversation (not reply to you when you choose a certain conversation path - leaving you stuck), merchants not opening the buy/sell window, horses that can't be ridden, and other interaction issues, then the save game is most likely corrupted. Try the SKSE ClearInvalidRegistrations console command fix I've posted above, instead.
To prevent this from happening in the PC, get the latest version of the Unofficial Skyrim Patch. This has been fixed since version 1.1 of the USKP:
"The Only Cure" (DA13) was never stopped which would lead to the Aspect of Peryite randomly initiating conversation and being unable to exit.
* This info is from the old USKP bug tracker page at 16bugs.com, which is now not viewable. The new USKP bug tracker report page of this bug is at afkmods.iguanadons.net.
Make sure that the mods' load order is correct (as explained in an answer by DouglasLeeder). If you're unsure on how to do this, just let BOSS take care of the mods' load order. BOSS has a database of mods and their proper load order, and can automatically decide optimal load order of mods, provided they are listed in that database.
Also, see the troubleshooting steps for the 'cannot absorb dragon souls' issue listed at the Unofficial Dragonborn Patch Nexus forums sticky post by Arthmoor, one of the UDGP developers:
Check for: dragonactorscript.pex and/or mqkilldragonscript.pex. Remove them if present. They are from dragon mods that came as loose files.
DSAMG - Dragon Soul Absorb More Glorious, and Diversified Dragons are known to cause this. Those mods need to be updated with a Dragonborn patch that incorporates the fixes from the UDBP.
Skyrim Unbound will cause this as well due to the script being unaware of the changes for Dragonborn.
Others may be a factor as well.
Note too that the offending mod may have the script packaged inside a BSA. That will need to be handled by that mod's author.
If you are using Mod Organizer and are here to report issues with dragon souls, sorry, but you're on your own as we do not support issues caused by incorrectly letting that program modify the BSA load order system the game has. Your post is likely to just be ignored. We don't have time to keep fending off false bug reports caused by people who insist on unpacking their BSA files using the program and thus subverting the entire system the game relies on for proper behavior.
...scripts are baked into saves (right?), so in theory I have to (cringes) restart the game to be absolutely sure...
Yes, scripts are baked into saves. However, there is a way to remove scripts left by uninstalled mods in savefiles. You need to configure Skyrim Script Extender (SKSE) to use its ClearInvalidRegistrations console command. It removes invalid scripts left running by uninstalled mods. This feature was introduced in v1.6.7 of SKSE:
add console command ClearInvalidRegistrations to remove invalid
OnUpdate() registrations
This prevents orphaned OnUpdate() events and
the resulting bloated/broken saves when removing certain mods. When
applied to an already bloated save, it will stop growing further and
instead shrink over time as the game processes all queued events. This
may take hours depending on the amount of bloat.
To execute automatically after each reload, add this to \Data\SKSE\skse.ini:
[General]
ClearInvalidRegistrations=1
Use the skse.ini method to automatically remove invalid scripts left by uninstalled mods.
I read somewhere that No Spinning Death overwrites a script related to dragon death. It might be mucking up things some, but I can't tell.
After configuring SKSE to use the ClearInvalidRegistrations command, try uninstalling the 'No Spinning Death' mod and then see if the issue is fixed. Otherwise, you'll have to disable half your mods and then test again. Rinse and repeat until you find the offending mod.
Best Answer
The crossbow in particular has had these issues since Dawnguard first came out, and they perpetuate in SE. The mods for first-person-only killmoves for classic Skyrim usually resolve the problem to some degree, but it won't fix misses. SE has no hope until we see SKSE for SE finished.
It seems to come from a "guess" made by the game when you fire a shot. If it would normally give a killmove for the shot, this triggers, but the game has to assume the kill or there'd be no move to display. That means, not calculating for falloff (since falloff for bolts and arrows in Skyrim is wildly unrealistic), your shot would probably have hit. Once the game imposed physics on the arrow, however, the shot actually misses.
This can happen in melee as well, if the weapon is particularly powerful or if something causes the target's ragdoll to react. This is most evident when using mods that break the enchantment rules; a 84 dmg dragonbone sword with 918dps/5 sec lightning will kill nearly everything in one swing, but if it's an opponent with immunity to shock damage, the killmove may happen only to allow the enemy to get up and fight back. This also happens in 3-party fights: if a Draugr Overlord uses Unrelenting Force or Disarm on your opponent while fighting them, you may enter a killmove only to be stuck because the target is no longer in front of you or has staggered below your swing, causing you to miss.