Sounds like it could be the generator, if your engine has one. If not, it is likely a "parasitic drain," meaning an open circuit which is conducting electricity when the mower is shut off. Usually these are areas where insulation has rubbed off a wire, or perhaps a safety switch which has failed and is allowing current to drain.

Your engine is a Briggs Vanguard. It does not have a generator. Rather it has an under-flywheel alternator with external regulator. The regulator p/n is 691185 which in turn has been replaced by 797375. Costs about $50 to $60.

Regulator is most common failure. Could also be bad winding in alternator. If you have digital voltmeter, you can test by fulling charging battery, starting engine and observing voltage a battery B4 starting, at idle, and at full throttle. Battery with engine off should read 12.5-12.7 volts. Will be similar, maybe slightly higher at idle. When turning full throttle (3300-3400rpm) voltage should be 13.5-14.5 if no difference, problem is charging system.

If you suspect current leakage, fully charge battery after mowing, and see if battery holds charge until next week. If you have digital meter, disconnect battery with everything off, and measure currently flow from battery post to dissconnected lead. 10ma or less should not cause problem from week to week. Anymore than that, and you need to trouble shoot.