The word unserer is not some kind of attributive adjective but in actual fact a possessive pronoun. That would be true for all case-inflected forms of unsere (nominative plural). However, as Canoonet states, these are often termed possessive articles rather than possessive pronouns in German, because if they are placed in front of a noun they act as an article.
Versuchen Sie unseren köstlichen Nachtisch.
Versuchen Sie den köstlichen Nachtisch.
Versuchen Sie einen köstlichen Nachtisch.
Schmeckt Ihnen unser Nachtisch?
Schmeckt Ihnen der Nachtisch?
As the examples above show, you would either use an article (both definite or indefinite) or a possessive pronoun.
But your example sentence and your detailed analysis show that there is both an article and a pronoun, so what is going on here?
The answer is that the sentence does not mean:
Try our delicious dessert.
(Versuchen Sie unseren köstlichen Nachtisch.)
A rigorous examination shows that unserer is actually in genitive case while einen remains accusative (as you suspected). The translation that is not the above but rather:
Try one of our delicious desserts.
(Versuchen Sie einen unserer köstlichen Nachtische.)
This also explains the plural form; had it been ‘Versuchen Sie unsere köstlichen Nachtische’ (accusative plural) you would have been expected to try all or at least many desserts – but common restaurant practice in Germany would have you order only one dessert per person so you can only try one.
The German construction at play here is essentially einer des Ganzen: a demonstrative article (einer) followed by a genitive of a plural or group noun. Essentially, it is a pick (any) one situation exactly like its direct English translation one of many. It is permissible to use einen von unseren Nachtischen, more closely mirroring the English one of, but using the genitive is usually considered better practice.