The cleanest setup is to decide what Houzz Pro is responsible for and what your accounting system is responsible for. If both systems are treated as the source of truth for everything, the numbers can drift.
For project management, I would want Houzz Pro to track estimate, proposal, selections, allowances, change orders, client approvals, invoices, payments, and project-level notes. For accounting, I would want the accounting system to handle bank feeds, payroll, taxes, reconciliations, vendor bills, chart of accounts, and financial statements.
The key is consistent categories. If your estimate uses one set of cost categories and your accounting system uses another, it becomes hard to compare estimated versus actual. I would map categories before the job starts: labor, subs, materials, fixtures, permits, design fees, freight, equipment, overhead, and change orders.
Also set a routine. Weekly job-cost review is better than end-of-project cleanup. Look at original budget, approved changes, actual costs, committed costs, pending costs, and remaining cost-to-complete.
Software can organize the information, but the process has to define who enters what, when, and where the final number lives.