• dāśārha — O descendant of Daśārha
  Daśārha is an ancestor of the Yadu line, also a name for the dynasty. Balarāma is addressed by clan-name — reminding Him that this is precisely the protective work His lineage is for.
• śuśrūṣaṇam — service
  Desiderative of √śru — literally 'wishing to hear,' but technically 'service' (the servant's first quality is listening). The sages frame the request as accepting their service of Him — the syntax is delicate.
• pūya-śoṇita — pus and blood
  Pūya (pus) + śoṇita (blood). Two of the six contaminants. The Bhāgavatam's surgical realism — it does not blanche from naming what Balvala actually does.
• surā — wine / intoxicating liquor
  Surā is the technical Vedic name for the prohibited intoxicant — opposed to soma. Balvala defiles yajña with the very substance Vedic ritual excludes.
• abhivarṣiṇam — raining down
  Abhi- + √vṛṣ (to rain). The same verb used in v.13-15 for the flowers showered on Kṛṣṇa. The asura performs an inverted varṣa — a downpour of impurities answering, by inversion, the deva-varṣa of flowers.

• Where am I trying to handle a gross disturbance with subtle tools?
• What firmer response does the situation actually deserve?

The sages list Balvala's pollutants in rhythmic dvandva — pus, blood, feces, urine, wine, flesh — and ask the Lord to receive their service in this form: kill him. The most direct service one can offer is naming the problem precisely and asking the Lord to remove it. Devotee-prayer is not bashful about specifics.