im not sure if it meets your requirements, but the Open Bench is 1/3 the price of the saleae and is open source.
The cons are less mature software and only 24kpoints or so, as opposed to millions for the Saleae Logic. The latter might be a significant advantage here, since it would allow you to sample at a reasonable sample rate for a long time. I don't think any affordable scope has the necessary amount of sample memory (say 300kS/s * 30s = 9Msamples).
Note that a logic analyzer will only distinguish between < logic threshold or > logic threshold, not sure if that's enough for this application. The input voltage is also usually about 5V max.
A DIY solution would probably be doable, but is likely to cost a lot more time if you don't have much experience, and costs may even exceed the $150 or so for a USB based logic analyzer.