Circuit doesn't work when multimeter/current meter is attached
Multimeters have something called a "burden voltage". See this PDF: uCurrent - A professional precision current ... - EEVblog.

Also see Dave Jones' video: µCurrent

Meters measure current by dropping voltage over an internal resistor, and measuring the drop. By definition therefore, there is a voltage drop, which is the "burden voltage". 


  the circuit misbehaves, never goes into sleep, and doesn't record any data


If you put a voltmeter on the processor side of the ammeter you will probably find that the voltage going into your processor is somewhat less than you think. That would account for the misbehaving.

One possible approach is to crank up the voltage from your power source, checking the voltmeter to see that it reaches the expected voltage on the processor side of the ammeter, however in a circuit which might turn on and off (ie. sleep) this voltage might jump around (depending on the current) so this should be done with caution (to not exceed the maximum processor voltage).




  Or do I need a different kind/quality of multimeter?


A higher quality meter may well help - watch that video and look at the PDF to see what I am talking about. My meters (which cost more than $15) don't have this problem to the same extent.