I would leave it as is for now...if anything is added I would suggest larger chunks of wood. Be patient and observant of how it fares.
You might poke under it a bit here and there, but nothing drastic.
Raking out and replacing would not be very frugal.
Nice run!
Lots of 'stuff' to 'do'.
Shavings do not make a true composting deep litter,
no matter how 'deep' they are piled,
time will tell how this bedding works at breaking down the poops.
Hopefully the birds keep it 'turned', keep and eye out for wet (anaerobic) matting.
Might want to put...
Post pics of your run, please.
Likely the ingredient you are missing is larger wood chippings(1/2" - 1" thick)...
....and maybe some moisture if you get no rain at all.
Grass, leaves, and shavings will clump obliterating any air pockets needed for decomposition.