Can I use lithium coin cell battery CR2320 for discharging about 80 mA at 3 V?
Read the datasheet.  It tells you what you need to know.  I could go into details, but that would be pointless as the datasheet already is doing a nice job.  You're complaint that it "doesn't tell me much" only shows you didn't really read it.

In particular, note the rated capacity of 135 mAh.  Even if the battery could deliver 70 mA and still maintain that capacity, how long would it last?  This is very simple arthimetic.  135 mAh / 70 mA = 1.9 h.  So it wouldn't even last for one of your 2 hour sessions.

However, 70 mA is waaaaay past any reasonable operating point for this battery, as the datasheet again makes quite clear.  Note that all the capacity ratings are with 10 k&Omega; (290 &micro;A) load, and that 600 &Omega; (6.8 mA) is considered a "pulsed" load.

As is too often the case here, the answer is READ THE DATASHEET.