You want to provide budget category and have the budget check enabled in expense reports, as it's available for purchase orders and manual supplier invoices.
SAP Business ByDesign 2108 Release
Budget category cannot be provided in expense reports and the budget check is also not enabled for expense reports.
However, you can check if you could make use of partner BADI BudgetCategoryDetermination to fill the budget category for journal entries of expense reports.
For budget check, ActiveBudgetControlUtilities.CheckBudgetAvailability API is made available which is also accessible for partners. Please check if this would work for you.
Help Document for SAP Business ByDesign 2108 Release: What's New in Financial Management > Section Active Budget Control for Public Sector Edition
budget control; budget category; budget check; expense report , KBA , SRD-FIN-ERM , Travel and Expense Management , How To