• cariṣye — I will perform
  Future 1sg of √car (to move, to enact). The same root that gives caritra, conduct. The Lord's first word in His response is the verb of voluntary action — He freely commits to the rite.
• vadha-nirveśam — atonement for killing
  Vadha (killing) + nirveśa (payment, expiation). The technical śāstric term for the prāyaścitta-rite assigned to a killer. The Lord names His act as vadha and accepts the formula's name for its consequence.
• loka-anugraha-kāmyayā — with desire to favor the people
  Loka + anugraha (favor, grace) + kāmyā (desire-driven). Kāmya-karma is action with desire — the Lord performs the rite as kāmya-karma, but His kāma is loka-anugraha. The desire-clause is the giveaway: He has no personal stake.
• niyamaḥ — regulation / rule
  Ni- (down) + yama (restraint). The procedure. The Lord asks the sages to prescribe the niyama for Him — submitting to their authority in form, even as He retains authority in substance.
• prathame kalpe — as the first option
  Prathama (first) + kalpa (option, alternative). Śāstric atonements come in tiered options; the first kalpa is the most rigorous. The Lord asks for the strictest tier — He will not cut Himself any slack on the prāyaścitta.

• Where could I place myself under another's guidance for the sake of those who watch?
• What humility does compassion require of me?

Cariṣye — 'I shall perform.' The Lord accepts the ritual. He who needs no purification freely submits to its motions, because compassion for lokas is the same word as compassion for forms. Balarāma's yes here is structurally identical to Kṛṣṇa's 'sarva-dharmān parityajya' — both presuppose that the One in charge of dharma freely chooses His relation to dharma's forms.