I was wondering why the "active meaning" and the transitivity of deponent perfect participles like cohortatus in (1) are not naturally preserved in the Ablative Absolute in (2). Why is it the case that the "active" use of the coniunctum participle in (1) sounds much more natural than in the absolute one in (2)? The construction in (2) was perhaps even ungrammatical for two great classical authors like Caesar and Cicero. As pointed out below in my first comment to TKR's answer, it seems that Sallust was the first author to venture expressions like Sulla omnia pollicito 'Sulla having promised everything' (B. Iug. 103.7)).
In any case, assuming the defining linguistic properties of deponent verbs, why should one expect that there is an acceptability contrast between the well-formedness of the participium coniunctum structure in (1) and the unnaturalness of the ablative absolute construction in (2)? In striking contrast, note that the present participle construction sounds natural in both syntactic contexts, coniunctum and absolute: Caesar, cohortans suos, ... and Caesare cohortante suos, .... Note also that the unnaturalness of (2), compared to the naturalness of (1), cannot be due to reasons of information structure.
(1) Caesar, cohortatus suos, proelium commisit (Caes. BG. I, 25).
(2) *Caesare cohortato suos, magnus timor hostes invasit.
When dealing with deponent verbs in Ablative Absolutes (AAs), the ones with passive lexical meaning like mori are, unsurprisingly, the prototypical/more frequent ones that enter into AAs (e.g., Caesare mortuo). Still, as pointed out to me by Joonas, deponent verbs with allegedly agentive lexical meaning like loqui can also be found in AA as well (e.g., locuto Caesare).
Furthermore, according to Oniga (2014: 308), "transitive deponent verbs, e.g., hortor 'to exhort' and arbitror 'to decide' (...) rarely appear in this kind of structure [sc. Ablative Absolute], cf. *hortatis militibus, *sententia arbitrata [* = ungrammatical: Mitomino])". I think that the "rarely" adverb in Oniga's quote is probably due to the attested non-deponent usage of these verbs (i.e., hortare and arbitrare).