1997 BMW 540i 1997 BMW 540i Engine fails to start after rep

Recently I let the car battery die from not driving the car for an extended period. I attempted to start it and it initially clicked but did not turn over. I then attempted to jump start it using my MINI connected direct to battery. It recharged it sufficiently to light everything up but failed to turn the engine or emit any sound at all. I then had the battery replaced by AAA service and everyhting lit up again but failed to turn the engine at all. AAA did apply a memory saver to the system prior to replacement. It is possible the power level was so low when I first tried to start it that it killed off any memory aspects it was retaining. Or perhaps my jump start fried something tho' I know everything was connected correctly. My call to the dealer said it could be the ignition or the starter but last time I started the car there was no indication of any abnormality in the engine starting tho I could tell the battery was not real strong at that time. So my question is: could it be the systems the dealer spoke about or is it possible something electronic such as a fuse or memeory module that needs to be reset or replaced? As I am unemployed I need to know which direction I am headed cost wise before I embark on spending money I do not have. Thanks.