That charging circuit is possibly only good for recharging what you'd use after starting & possibly lights. How fast does it kill it? Would think you could run quite a while on just the battery, even without a charge.
Is this alternator link available ?I'd double check that the replacement engine has the correct alternator stator for your system. Briggs has several different ones with differing outputs. Charge can also depend on the number and size of magnets on the inside of the flywheel. B&S alternators can be identified by the connector coming from the engine. I don't know off hand which one you should have but with a little research you should be able to conclude this. Try this download link and see if it could help you.
Briggs Alternator ID, & Wiring diagrams - Manuals - Garden Tractor Talk - Forums
Try this link - Briggs Alternator ID, & Wiring diagramsIs this alternator link available ?