You don't need to enclose the testbed, so I'd recommend turning on easy mode - Pick up a uCurrent or two.

http://eevblog.myshopify.com/products/ucurrent

There's an internal battery, so it won't affect your measurements (or hook it up to an external benchtop power supply to run for months)

I have though about it, but I don't really know how to measure current that vary from a few μA to 100 mA with the Arduino (ie the difference between sleeping and waking SD cards).  There is this post over at Dorkbot: http://www.dorkbotpdx.org/blog/paul/measuring_microamps_milliamps_at_3_mhz_bandwidth

but building a logger based on that method will require a chunk of time. Are there any nice I2C μA range sensor boards out there that would let me cobble something together quickly?

I'd recommend building a test logger that does nothing but measure the current draws of all of the devices and logs them to the SD card. Assuming you get similar anomalous data from the test rig, you should have complementary data that characterises the current draw from the SD card, Arduino, overall current being drawn from the battery, etc.