This is a trickier question than I thought at first.
If they were both departures, and assuming the intersecting runways diverge by at least 15º, the answer would be obvious: All you need is "through the intersection," reference 5-8-3d2. And if the two runways were non-intersecting you would have no minimum separation and could launch the departure even with the arrival on short final, reference 5-8-5c. But in this situation neither of those apply.
you would need to provide standard IFR separation once the departure is airborne.
If the runways diverge by at least 15º, you are allowed (5-5-1b2) to use passing-or-diverging (5-5-7a1(a)), which means you have separation as soon as the departure crosses the intersection. Just make sure not to give a boneheaded vector to the departure.
The problem is:
What separation are you using if the departure rotates before the intersection, assuming tower visual isn't an option? And even if tower visual is an option, what separation are you providing before the application of tower visual? 3-10-4a1 is well and good for VFR, but the segment of time we're concerned about is that time between when the departure rotates and when they get through the intersection. I could see an argument for applying 3-10-4 prior to applying tower visual, sort of, but it doesn't make me very comfortable.
(I also almost made the argument that the wording in 5-8-4 implies you need to provide IFR separation even before the departure rotates, but then I realized that 5-8-4 is nothing but a waiver allowing you to drop below 3 miles of airborne separation. The "separate by 2 miles" thing is just that the condition for that waiver.)
The problem with trying to use 5-8-4 is the Note, which reads:
This procedure permits a departing aircraft to be released so long as an arriving aircraft is no closer than 2 miles from the runway at the time. This separation is determined at the time the departing aircraft commences takeoff roll.
Now what does "the runway" refer to? The simplest explanation is that it refers to the single runway that is being used by both aircraft, which would mean this rule cannot be applied between two aircraft on two different runways. But you could sort of squint your eyes and pretend that it refers to the arrival's runway, and I can see that making sense. In this interpretation you would be allowed to launch the departure so long as the arrival is no closer than a 2-mile final.
But that's just the Note, not the paragraph itself. The paragraph doesn't mention a runway—it just says "separate... by a minimum of 2 miles," which you could interpret as the straight-line distance between the two aircraft at the time the departure starts rolling. Depending on the layout of the airport you might have that even when the arrival is over the numbers, so that's probably a poor interpretation.
So in the final analysis, and without considering the complication of tower visual, I offer two possible interpretations:
- Squint your eyes and apply 5-8-4. The departure has to be rolling by the time the arrival reaches a 2-mile final.
- Apply standard 3 miles. You need to have 3 miles between the two aircraft from the moment the departure rotates until the departure crosses the intersection.
That second option would be even more separation that you implied in your question: Not only 3 miles between the arrival and the intersection when you launch the departure, but 3 miles between the arrival and the departure
at all times, meaning you would need the departure to be rolling before the arrival was on a 4-5 mile final, depending on speeds.