Can simultaneous departures be authorized from converging airports?
If this were in an FAA ARTCC's airspace, they'd handle them as one airport, because of their proximity.
The protected lateral airspace for an aircraft is 4 miles either side of the course to be flown. That eliminates a lot of possibilities.
In a non-radar environment, with only 3.8 miles between the airfields, you need a reduced separation rule, such as the 13/3 DME rule (7110.65 6-2-1, a3). One and two minute rules won't work, because of the convergence.
If you have a NAVAID that the departures fly over, you can use 3 miles between aircraft (faster in front), until their courses diverge.
The divergenge has to be 45 degrees or more, and the NAVAID has to be within 13 DME of both airports to  use the rule.
The 44 knot rule (7110.65 6-4-2) could possibly be used if you can maintain a minimum of five miles between the aircraft. Remember to account for any local speed restrictions (such as 250 knots below 10,000 feet) that may keep you from having 44 knots between your aircraft. Also, the 44 knot rule is based on filed true airspeed, not ground speed. But again, someone has to wait.
If this is a radar environment (I'm assuming you use 3 miles separation), you can release the second aircraft as soon as you get radar on the first (using radar contact within one mile of runway end) as long as you maintain 3 miles.
These rules are usable in either terminal or enroute environments. There may be more in 7110.65 chapter 3, but I don't go there, being a center controller.
If it's possible to post a plan view of the airspace, there's some CTO rated controllers on here that could give more definitive answers.