tophat size recommendation

I have a section of old ceiling that is 2.2 wide and maybe 10m long. It used to have 3 walls holding it up, now none. What's left is the top plates, and above the old baltic lining is 3 hardwood joists running lengthways that look like they started as 5x2s, so now about 125/130 x 50.

I need to plaster it. I'd like to use a tophat batten that runs under the old ceiling level that will provide support for the skillion roof above by fixing it to the walls (and a beam) that I can then fix the gyprock to.

What size tophat or other style batten would do both tasks? I'd prefer the lowest profile possible to limit the reduction in ceiling height.


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