If these were Polish, they wouldn´t be unusual enough to call them exceptions

OK No objections to your calling it exceptions but possessive exceptions.

I don´t see the possessive suffixes functioning irregularly with those you call exceptions. (I have provided some examples of possessive suffixes functioning irregularly.)

There are some irregularities with those words that happen when any (not just possessive) suffix is added to them.

Does ömür behave differently when other suffixes are added? As I don´t know.

It is " Orta Hece Ünlüsü Düşümü " middle syllable vowel dropping.  Whether you call it as an exception or a fonetic development, does not change the fact that  there is " vowel dropping"  happening with some words or .  It is not seen only with possessives, it can be seen with other suffixes. For instance , take one of those word:

İzin  + e  ---> izne   [ to day [s] off  from work ]
Yakında izne çıkacağım  ---> I will have days off from work soon. [ in other words, I will have get holiday from work]

Middle syllable dropping, because, when it takes a suffix, it becomes the middle syllable.

izin

İ + zi + ne    ----> izne

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ömür
ö + mü + re  ---> ömre

Seninle bir dakika ömre bedel ! ---> Being with you one minute , worth a life !

* However, when the suffix starts with a consonant, then the vowel dropping don´t occur.

Ömür + den  ---> Ömürden
Ömür  + de ---> Ömürde
Ömür + süz   ---> Ömürsüz
Ömür + lü ---> Ömürlü
Ömür + se  ---> Ömürse