My first path is discerning how he is related to the artifact and not his absence in during the aforementioned periods.
At first I thought that perhaps Peryite would favor the Dwemer or more specifically the Rourken clan of Dwemer, the original makers of the shield. However this idea has two major flaws. One, the Dwemer's only connection to anything beyond the plane of Mundus is the Heart of Lorkhan and have a well established disdain for daedra. The other being that the Dwemer tampered with the Heart and potentially could have damaged the natural order of things, something that would anger Peryite. So I find it hardly likely that Peryites connection lies in some sort of amicable relations with the Rourken clan, much less any other Dwemer.
Next I thought that the connection may lie in the nature of the Dwemer's disappearance. Of course nothing is known for certain other then their dissappearance after using the Heart, one may be able to surmise that Peryite took possesion of the artifact perhaps due to its ability to alter reality, limited in scope as it may be. Considering the shield was fabled to stave off magical attacks, Peryite may have seen this as a potential threat to the order of things and wanted to safe guard it. This theory does have the chronological benefit as the Dwemer dissappeared 1E 700 and its first appearance outside of the battle of Rourken-Shalidor was when the Enternal Champion acquired it around the year 3E 399 (or perhaps 3E 389). My only difficulty with this notion is that it would only effect Nirn and Peryite is the Taskmaster of the lowest planes of Oblivion and not the mortal planes.
My last thought on how it may be related to Peryite is in the enchanting process itself and how the process may be related to Peryite. It's obvious that the shield is of Dwemer make however it may be possible that an entity assisted in its enchantment. As previously stated the shield has historical implications in the Battle of Rourken-Shalidor. Not much is known about the battle, however Shalidor was the Archmage who founded the College of Winterhold in the first era. It can be extrapolated that the Rourken clan and Shalidor had relations, whether they were amicable or otherwise. Its unknown what type of relations they may have had or what led to the battle however the enchantment placed on Spell Breaker is reminiscent of Shalidor's Mirror, a powerful spell reflection spell. So it is possible that the shield itself was created both by the Rourken clan and Shalidor, perhaps in a vain attempt at peace. This extraplotion leads me to Peryite's involvemnt in the enchantment process. Shalidor's trafficking with daedra is not documented however he may have had dealings with Peryite and used his assistance in the enchanting of the shield. This would explain Peryite's connection and why the shield is of Dwemer make.
Of course that is just my theory.
Now for Peryites absence of its appearance in certain cases. In the case of the Nerevarine's ability to acquire it, it is well known that many artifacts 'have a mind of their own' and travel from person to person, era to era. Although I see this as a means for the continued existence of powerful items throughout the series it could be supported both by Daedric involvment and the souls used in the enchanting process. As I've mentioned the only credible creation to Spellbreaker came from Peryite's assistance, it could be said that Peryite controls its movements either directly or indirectly. Peryite, as many other entities, would have viewed the devastion of the Blight as an unnatural abhorrence. The Blight itself is not an actual disease but rather a bit of divine power affecting an individual and altering their state of mind and physcial attributes. The Blight, or Divine Disease, was influenced by the Heart of Lorkhan and Peryite may or may not have seen this unnatural balance as another thing to correct. Peryite may have simply allowed Spellbreaker to fall into the Nerevarines lap in hopes of adding him/her in stopping the Blight and allowing the land to return to normal. Again this falls into the category of Peryites charge being daedric planes and not mortal ones.
So... yeah... Thats what I found out, any thoughts?
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Referrences
http://www.uesp.net/wiki/Lore:A_Minor_Maze
http://www.uesp.net/wiki/Morrowind:Dwemer
http://elderscrolls.wikia.com/wiki/Spellbreaker
http://www.imperial-library.info/content/shalidor-archmage