Running 1.16.5 0.3.3 (modpack Crazy Craft Updated), I tried normal axe and a silk touch axe (I even busted one using the chicken stick just on the off chance that worked like a wrench). Every time I got back an Empty Roost, tried with both tier 1 and tier 2 roosts. The only way I was able to move a roost after placement was using a cardboard box from Mekanism. (I'm guessing Carry On or something like that might work too, just don't have that in my current mod pack.)
Running 1.16.5 0.3.3 (modpack Crazy Craft Updated), I tried normal axe and a silk touch axe (I even busted one using the chicken stick just on the off chance that worked like a wrench). Every time I got back an Empty Roost, tried with both tier 1 and tier 2 roosts. The only way I was able to move a roost after placement was using a cardboard box from Mekanism. (I'm guessing Carry On or something like that might work too, just don't have that in my current mod pack.)