I would take the cage into a small room that a door can be closed. A bathroom can work , but I would tape a sheet over the mirror, close the toilet and put a towel in the bottom of the tub. If it's a glad shower you would also want to cover the glass. Thst way if a bird escapes it won't crash into the mirror or glass. They are contained in small room so it's easier to catch them . You can try taking things out and cleaning them and putting them back .
I think it's easier to remove the birds , but it can be traumatic for you and stressful for them. What I think helps a bunch , is once you have them in the small room, and have the place ready to transfer them to ready , you can use a box with little holes and a towel on the bottom with a perch laid on the bottom . Or even a large plastic tub with same prep . Then careful reach in and remove some perches and food dishes and things they might flap into from their cage if possible. Then turn out the lights and have it decently dim . Give it a couple of minutes. Then reach into their cage and grab them one by one and transfer to your temporary set up. They might nip or protest . Then you can thoroughly clean their cage . I use dawn dish soap , I use dilute vinegar in cold water to help break down stuck on poop.