Hello,
I would interpret the "recommended discharge current" to represent the average continuous current level, to achieve, in this case, a battery life of about 1000 hours. It is probably the current level used to rate the mAh capacity of the battery. Current peaks of short duration may exceed this value, limited by the internal resistance of the battery, particularly near the end of life of the battery.
Assuming the instantaneous load is 50-60ma, and with a voltage reduction of 1.2 volts, this would indicate internal resistance of about 20 ohms. This seem to suggest that the battery you have been using is exhausted. Perhaps you might measure the reduction of voltage with a fresh battery and a known load to get some idea of the allowable peak current for the battery. It will also depend on the minimum voltage your project can tolerate, and what battery life you require.
If the CR2032 coin cell does prove to be inadequate, you might consider a larger Lithium coin cell such as CR2450. Another possibility might be to use two CR2032 cells connected in parallel.
Regards,
Mac