Your tractor is plenty big for the JD350. I had a 9' 350 on a Kubota L2900 and it works okay; I would not want to use a smaller tractor, but your L39 is much larger. The issue is not power, but weight. I now run the 9' on a Kubota B2400 with 18 pto hp and it does not even load the engine. But the heavy bar and wobble head are offset a foot or two beyond the right hand lift arm of your TPH and it takes the substantial tph of a 2,500 to 3,000 pound or more tractor to manage the offset weight. When I moved mine to the B2400 I built a frame that hooks to the rear axle housings and FEL brackets (like a backhoe frame) to hold the mower because the little B2400 tph was clearly too lightly built.
I agree with Mark; the wobble head can be very expensive to repair. To check it out, let it run slowly while you pour lots of oil on the knife (the thing that holds the triangular cutters or "sections") so they are quiet for a few seconds. The knocking may just be the 1" or so diameter drive pin that connects the reciprocating arm in the wobble head to the knife head; that pin and the needle bearings in the knife head are not too costly ($75). You can tell if the pin is loose by slowly turning the flywheel back and forth a few degrees while you place your fingers along the joint between the reciprocating arm and the knife head; if you feel motion the pin is worn. But if the knocking comes from inside the wobble head you will be lucky if it is only the bearings and not the several hundred Dollar crankshaft or related parts; I probably would not buy it with such noise.
It will cut the brush along the sides of your driveway as you mention, but you will have to rig up some linkage to hold the cutterbar vertical and to keep the whole wobble head and cutter bar assembly from bouncing up and down on the spring. Mine will cut 1" brush if you go slow, but I try to limit that sort of cutting because it dulls the sections and the noise from the machine suggests it doesn't like it.
I bought a second JD 350 with a 7" cutterbar that I am rigging up to put on the fel of a 40+ hp tractor with a hydraulic drive to cut banks and the sides of driveways. But that project is just in the planning stage.
I think I paid about $1,800 for each 350 and they turned out to be in pretty good mechanical shape. After a year I replaced the knife and rock guards (the pointy things that the knife sections reciprocate in to cut the grass) with a complete system from Schumacher that is used on many grain combines. It has roller guides rather that wear plates and hold downs, and I am very satisfied with it.
Let us know how it turns out.