Do they turn off their 2/3/4G services at the tower, or do they just stop passing traffic to their upstream providers?
If the latter then it may mean that devices on the same network can still communicate with each other. I don’t think this will solve your problem, but it may still be helpful in some cases.
Ethiopia only has one AS/provider apparently: 24757 ETHIO-TELECOM
During one of the many internet lockdown I've experienced there I know Ethiotel allowed some Embassies to have their ADSL restored (the lockdown lasted almost a month): there was no internet even in very expensive western hotel: a couple of manager where telling me they were about have the internet back, but then this did not happen.
As for internet on mobile (which, btw, has becoming really fast last year: 4G with easy 100Mb up/down, in central Addis only, of course) I expect some people close to the government may still have internet access (the country have strong socialist background, and is not Nigeria, but power and money can move a lot as everywhere).
Technically I think it should be pretty easy to selectively lock the internet by working on the APN: you can lock the ordinary one that is pushed on every phone, and have another one for "friends".