Hello,
This type of problem is called a parasitic draw. That is when there is a module or a short like you suggested that is causing a drain on the battery when it is sitting. I thing the battery replacement was a good guess. This vehicle is not common for getting parasitic draws actually but it is possible. We see problems with general modules causing a drain or lamp modules at times. To really find your cause for the battery drain a parasitic draw test will need to be performed. This is hooking an digital volt ohm meter in series with the negative terminal looking at amperage. After an half hour the vehicle assumes a "sleep"mode and only should be pulling 30-50 milliamps. How we find the culprit causing the battery drain at the dealer is to then monitor the drain...say it is 1 amp...and then start pulling fuses and disconnecting modules 1 by 1 unil the meter drops down to a normal reading of 30 milliamps. Then the fuse diagram is checked to see possible modules or if it was a specific module then that one would be faulted out and would need replacement. I have also seen mice damage to wiring cause a drain as they seem to like Z3's for some reason. I actually found a nest behind the glove box one time with chewed wires causing the draw.
You should also have the alternator checked to make sure it is putting out the correct chargning voltage as I have seen intermittent alternator problems when it is starting to fail it will charge sparadically. Most often though this would show as a battery warning light but it would be worth checking as well.
BMW Mechanic
Level 1 BMW Master Certification, 9 Years Dealer experience, Team Leader, Shop Foreman