My BX2200 should be similar. I used my solvent tank, which is filled with Naptha.
This is flammable. If yours is attached to the tractor with what looks like a large bolt, then all the components are metal except for two large o-rings. Protect the O-rings from solvent, and clean the rest with any appropriate solvent, flammable or not. Just don't breath it, or blow yourself up somehow.
The large bolt head attachment style is not suitable for a quick clean using a hand made plug. With the strainer removed, there is no pressure to keep the back side o-ring seated, and the entire banjo fitting attached to the hose is free to move around, unsealing the o-ring between the banjo fitting and the transmission.
I would not attempt to remove the strainer from the bolt head thingy. There is not need to do so, and your are really breaking it if you remove it from the bolt head thingy. (I wish I knew what this is called.) Oh...it's all called "Filter"