I would check the battery terminals, probably a buildup of corrosion between the posts and the terminals, clean them up with boiling water and sandpaper till both parts are shiny clean. If you are connecting the jumper leads to the battery terminals when jump starting and it works fine then this is the only possibility.
As long as the battery in the lawnmower is a 12 Volt battery you will be fine.
you need a new battery yours does not hold its charge for the cranking amps needed or you have a dead cell in it.
Does it crank over fine?
Battery
If you can jump start the car then the battery is not fine. Remove the battery cables and thoroughly clean them. If it still will not start check the voltage at the battery with the engine off. You should get 12.6 volts if the battery is fully charged. 12.4 it is 75%, 12.2 it is 50%, and 12.0 it is 25% charged. I would bet your battery is defective.
A clear sign is if your vehicle wont start without a boost, but fires up fine when boosted. Also dimming lights when vehicle is shut off or running, all signs you alternator is not charging your battery.
when you unhook the battery the computer probably resets itself.
Bad solenoid, bad ground, faulty ig. switch.
Have you removed the battery cables and cleaned the battery posts and cable connections? This is a frequent cause for the problem that you describe.
If alternator and battery test fine- then you may have a bad fusable link not allowing the alternator to charge the battery
No, it will be fine. Just install a battery and fire it up.
Check for loose or corroded battery cables Could be alternator is not fully charging battery - check with a voltmeter A fully charged battery should read approx. 12.8 volts without engine running - should read approx 13.8-14.2 with engine running Could be a bad starter or starter solenoid