Where are you connecting the booster battery? If you are connecting it to the battery terminals, and it works, then the problem is that the on-board battery cannot deliver enough current. The problem must therefore be loose connections, discharged battery, or faulty battery.
If you are connecting the booster elsewhere, then the problem is in the wiring between the on-board battery, and wherever you connected the booster.
Tecumseh engines have more reliable electric starters than the equvalent Briggs - the Briggs engines need a FULL battery charge, as the geared starters that these engines use only just generate enough RPM to start the engine - even when new. The Tecumseh arrangement is more like a car, and generally works better.
phil_saunders(a)bigfoot.com
Phil Saunders, April 2010