When a Faction per the Event Card sequence of play (2.3) executes
an Operation in at least 1 space (3.0), it may also execute 1 type of
its Special Activities (exception: Limited Operations, 2.3.5).
So you have to execute an operation in 1 space in order to be able to carry out an SA. So unless you can select a space for an operation and spend zero resources, you can't carry out the operation.
The Coalition could actually do this by selecting a space for Train and then not adding any cubes since they only pay for a Train if they add cubes.
If Eradicate added resources directly (instead of aid/patronage), you could do the SA first, get the resources, and then execute/pay for your operation. Since none of the Gov't's SAs adds resources, you won't be able to conduct an operation.