I don't intend to count votes-by-proxy from discord. If anyone ever puts another person's name on the wrong vote it's going to be way too much effort to sort out what they really wanted to vote. Given that it's a ghost town in here I'll take two votes though.
Roll: 4
Difficulty:
This design originally called for an elaborate set of gears which translate rotation in a single direction to rotation back and forth. In fact, such a gear attaching the wheel to the action isn't necessary at all. A much simpler connecting rod can translate the rotation of the wheel into a reciprocating motion, like the way in which pistons push the wheels on a steam locomotive. This is basically how the original gun works already. The actual question is how to make the wheel move in the first place. The original wheel had a piece which was pushed away from the barrel by gas, and then returned in time to be pushed away again. The need for mechanical advantage for this escaping gas, and the need to have the reciprocating motion travel a certain length, are factors which tend to make the wheel larger.
Your engineers are stumped on this problem for a while, before building a prototype with a bicycle freewheel- a ratchet, basically. This lets the flywheel be a different size than whatever the gas port is pushing on. Since minimizing weight and keeping the gun steady are the main priorities, you settle on a very small and short lever which springs closed. This necessarily means less mechanical advantage, but the energy stored in the flywheel should help make up for that. Essentially you have redesigned the old "potato digger" machine gun design with a much small lever that doesn't buck the gun around or strike the ground, and made up for the lack of leverage with the flywheel.
After using most of the schedule on this novel mechanism, there's precious little time for weight savings. Besides making the barrel and tripod lighter, holes are cut in the wooden furniture. It's still a very heavy gun. Regardless of size, the flywheel is also more effective at storing energy the heavier it is, so it weighs more than a kilo. 100-round belts are trivial to make available, but you don't figure out a good way to attach wooden ammo boxes to the gun without adding a lot of weight and complexity, so that idea is scrapped. The four barrel fins are of dubious value, so they're dropped for cost and weight.
FR No.2.b Light Machine Gun "Weil": This shares most of the rear action with the original Weil, but the entire gas mechanism is vastly different. A gas port in the barrel pushes a small lever, which exerts force on a spinning flywheel in the position of the old wheel through a ratcheting mechanism, like a bicycle's freewheel. This flywheel has a coupling rod to the reciprocating action of the gun. The trigger disconnects this. It can be spun up with a crank, and then maintains speed for up to 30 seconds before coming to a stop. Each time the gun fires it accelerates the flywheel a little bit. The whole time the flywheel is spinning, it makes a distinctive ratchet noise. The exact rate of fire depends on the speed of the flywheel, it can operate as low as 90 rounds per minute, and will accelerate as the gun continues firing, up to 500 rounds per minute, at which speed the gas lever is not returning for every shot so the gun loses a little speed. The top rate of fire may be much lower depending on how well-lubricated the gun is. The gun will fail to chamber a second round if it is fired when the wheel is at a complete standstill, and it often fires once after the trigger is released. There are also some weight saving measures: the barrel is shorter, lighter, has no fins, and lacks a quick disconnect mechanism. The removable wooden stock has a large hole in the middle, and the tripod is smaller and lighter. It's portable by one man, although someone else will still have to carry the ammo. 42 credits per batch.