Is it permissible to calculate a separate hawl (Zakat year) for future installments and payments from new money while only paying Zakat on the remaining balance of it, or must all funds be combined into one pool and Zakat paid on them together?

There is no harm in delaying the zakat (definition) on the second sum of money until its (hawl) year passes. Debt is deducted from the zakatable wealth, considering all that the zakat payer owns in excess of their basic needs. When paying zakat on the first sum of money, the debt is offset against both sums combined, and the entire first sum is zakatable. When the year passes on the second sum of money, both sums are considered as one wealth, and the amount corresponding to the debt is deducted from them, then what remains of what has completed its year is zakatable.