What is the most idiomatic verb to use with ‘reading’?

Conduct is fine; other verbs are possible depending on context.

Your usage of reading is accepted jargon within literary criticism. It generally means an analysis of one or several texts, or an analysis of a phenomenon using texts. (See close reading [M-W], which has led to analyses based in other theories,e.g. feminist reading, deconstructive reading, and ecological reading.)

In my experience, verbs that would be used with an analysis would also work well for doing a reading, like conduct. A verb-based collocation search with "analysis" in the Corpus of Contemporary American English confirms this hunch; the top results are

• perform and performed (111 and 96 results)
• conduct and conducted (84 and 95 results)

Indeed, in searching for actual usages with reading, conduct and other verbs are used:

How do we conduct an ecological reading of premodern cultures without being merely presentist, polemical, or romanticist? (Ritual Dynamics in Jewish and Christian Contexts: Between Bible and Liturgy)

Ruth Mas conducts a close reading of the concept of translation itself, as Mandir develops it mainly in relation to Derrida and to Nakoi Sakai, who develops his idea of co-configuration in relation to Kant's idea of schematism ("The Past Future of Postcolonial 'Category' Religion, Theory, and Translation")

The data of biblical theology, in both the Old and New Testaments, allow us - if they do not indubitably require us - to develop an ecological reading of biblical faith, focusing on the themes now familiar to us, especially in the works of Irenaeus and Augustine, as integrating constructs: the land of promise and the fecundity of God. (The Travail of Nature: The Ambiguous Ecological Promise of Christian Theology)

In one of these assignments, students perform a close reading of passages from Orientalism (Said 1978) [...] ("Rethinking Combined Departments: An Argument for History and Anthropology")