The first sketch shows the problem (stick smack dab in the middle) and the solution:
A spring-loaded hinged armrest with a cheap but effective screen-door latch holding it down.

To release it, all I'll have to do is reach under and depress the little handle (and get my arm out of the way). Might rig a cable release someday, but this will work just fine for the moment.

Might have to actually pay for the right spring, or maybe a bungee... hmmm, a bungee would be quieter...and cheaper. Much cheaper.
I have a couple of 45-lb gas springs, but even one of them would be too much, unless I made the armrest out of cast iron.

Open... and closed. Simple, and it will reveal a little cubby for charts, etc.
Two birds with one stone... I love it when that happens!

Of course, old Archimedes might have to intervene, here... I may need to have to set the spring back a ways more, to provide enough leverage without having to have the spring so tight that this thing whips up too quickly. Thge lighter the armrest, the less need there will be for that- but I can't make it too flimsy, as it will be abused more than any other part of the simpit.
But I've got a foot or so beyond the fulcrum, so it'll work one way or the other.

The cables from the stick(not shown here) will most likely run out a hole in the end and go down alongside the spring or bungee or whatever...
Well, concept-wise, it's all downhill from here... pretty much just have to get down to it and finish building this sucker.