• vidhi-vat — in accordance with scriptural injunctions
  Compound: vidhi (rule, scripture) + vat (adverbial suffix). Vaiṣṇava bhakti is not anti-scriptural sentimentality — vidhi-mārga structures sādhana until rāga-mārga blooms. Hari-bhakti-vilāsa codifies precisely this.
• kṛṣṇe naḥ bhaktiḥ astu — may we have devotion to Kṛṣṇa
  The Vṛṣṇis' explicit prayer. This single phrase is the Gauḍīya prayojana — Caitanya Mahāprabhu's prayer in the Śikṣāṣṭakam (mama janmani janmanīśvare bhavatād bhaktir ahaitukī tvayi) is its direct descendant.
• bhaktiḥ — devotion
  From √bhaj — to share, to participate, to serve. For Gauḍīyas, bhakti is both means and end — sādhana-bhakti ripens into bhāva-bhakti, then prema-bhakti; it is never instrumental to liberation, because it surpasses liberation.
• dvija-agryebhyaḥ — to first-class brāhmaṇas
  Compound: dvija (twice-born) + agryebhyaḥ (to the foremost, dative). 'First-class' in Vaiṣṇava reading means brāhmaṇa-vaiṣṇava — those whose dvija status is completed by bhakti. Mere birth is not sufficient (cf. SB 7.9.10).

• Is 'grant me devotion' a regular prayer of mine?
• What would change if it became the capstone of my practices?

A merchant offered every commodity at a fair will pass them all by and ask only for one rare gem. The Vṛṣṇis bathe at famous lakes, feed first-class brāhmaṇas — and then ask for the one thing those acts cannot produce on their own: kṛṣṇa-bhakti.