Hold on! The Engine Computer is only one of your worries. There are other computers including the Security system and the radio that will lock up. If you have the famously unreliable and dangerous throttle-by-wire system that put Toyota in the news, (and will again), that system will lock up too. You can introduce a whole pile of expensive problems by just disconnecting the battery or running it dead. I can't speak directly to your car, but Volksawagen started this misery in the '90s and it's a common problem now on most car brands. Nothing about this nonsense benefits the owner. It is a means of the manufacturers bleeding more money from their customers.
If you want to replace the battery yourself, use some type of memory saver device. There are units that use a 9 volt transistor battery and plug into the cigarette light or power outlet, but those will only work if those sockets stay live without turning on the ignition switch. I use a very small battery charger set to its lowest setting. Connect the ground clamp to the engine block where it won't be disturbed and clip the positive cable to the stud in the under-hood fuse box or remote jump-start terminal. When the positive cable is off the battery, be very careful that it doesn't touch anything metal. Also double-check that you aren't connecting the cables backwards to the new battery. I read about that a lot here. If you're lucky you'll just pop a bunch of fuses, but it's also possible to damage numerous computers.
The manufacturers have taken this simple task of replacing a battery and turned it into a nightmare worthy of a nervous breakdown. Even if you have a car such as a Chrysler where the computers and radio don't lock up, it's possible for multiple fuses to blow when you connect the new battery if you don't use a memory saver device. That's due to the current surges when the various computer memory circuits charge up when the new battery is connected.