You could flush the radiator with citric acid or another good product like Barnacle Buster leaving the radiator in place, by hooking up hoses to the top and bottom connections, then running those hoses:
(1) one hose to an immersion pump you place in a large bucket filled with the necessary liquid (first water to flush out coolant, then the cleaning agent, then water again to remove the cleaning agent before refilling the system with coolant)
(2) the other one back into the bucket so the pump stays immersed as it is doing its thing
The pump will circulate the liquids (water/cleaner/water) to circulate through the radiator however long you want it to. Once the cleaning agent is in, you can let it sit for a couple hours if you like, then start up the pump again. I used a small, simple 12 volt bilge pump widely available powered by a car battery both times I did this procedure. You can also use a household current immersion pump (like used in basements or for sprinkler systems), but for me the 12 volt pump was plenty strong on car radiators. Below picture is of my set-up on my Lancia Fulvia radiator (which I had out of the car, but to get the basic idea)
(When I did this procedure on our boat's massive Caterpillar V8 turbodiesel engines I did both engines at the same time, connected together, using a 110 volt pump).