You should use Konjunktiv/Subjunctive I here. The infinitive clause contains an example of indirect speech (which extends to thoughts and beliefs). Indirectness is a typical domain of the subjunctive mood. In written German, such an indirectness subjunctive is generally expressed using the Subjunctive I provided that the subjunctive form of the verb is unambiguous, in the sense that it is clearly recognisable as a subjunctive. This is the case with (sie) habe.
That being said, a more thorough analysis of the subjunctive mood will reveal that there are contexts where Subjunctive II is used by speakers in indirectness contexts. Keep in mind, though, that exploring these uses is probably not suitable for a German course/exam. Quite a few people will consider all or some of these uses as non-standard and, above all, there is a lack of clear rules (see Fabricius-Hansen et al, Der Konjunktiv, 2018, 162ff, for a comprehensive survey).(*) In many cases, it simply seems to be down to individual preference by certain speakers/authors (Duden-Grammatik, 9th edn 2016, para 777).
If you convert the infinitive clause into a dass clause, that does not change the above. Note that switching to a dass clause opens up an opportunity for omitting the subjunctive mood altogether (ie using hat instead of habe). As you may know, in spoken German, the Subjunctive I is generally disfavoured, so if you listen to someone speak German, you should always expect to hear indicative rather than Subjunctive I. But in this case, I would personally be inclinded to use indicative mood even in written German, presumably due to a culmination of known factors that favour indicative over subjunctive use in indirectness contexts (indirectness context already clearly indicated in the matrix clause (Überzeugung); subjunctive unnecessary for understanding the temportal structure because the matrix clause clarifies that the anteriority of the reported fact is in reference to both the figure's now and the speaker's now; etc). See generally Zifonun et al, IDS-Grammatik, 1997, vol 3, 1766ff; Fabricius-Hansen id, ch 5. However, if you prefer to err on the side of caution, you should stick to the Subjunctive I, which is universally accepted.
(*) For the linguistically inclined, which or may not include the person who asked this question: There is, for instance, a view that the Subjunctive II is used in lieu of Subjunctive I in indirectness contexts to additionally confer a certain degree of irrealis from the speaker's point of view (somewhat controversial, the opposing view being that it's either an indirectness marker or an irrealis marker, but not both). Applied to our example, the idea could be that the speaker might wish to not only report, but also cast doubt on the veracity of the figure's conviction. See eg Jäger, Der Konjunktiv in der deutschen Sprache der Gegenwart, 1978, 130ff, 158ff. Such a 'universal' role of the Subjunctive II is disputed by others (eg Fabricius-Hansen id, 157ff). However, I suppose the user who posted this question produced the sentence on their own and probably didn't intend for that anyway.