Removal of the laser gantry might not be worth the effort for a cleaning. They went out of their way to discourage tinkering by designing it not to be disassembled.
The black coolants tubes can be clipped off using heavy duty paper binder clips. As long as you don’t lose too much coolant (say under a tablespoon or so) you should be OK. Any air bubbles will work themselves out very quickly then next time you power it up.
The bigger problems are the high voltage feed (the red wire connected to the right side of the tube) and return (the black wire connected to the left side of the tube). These can only be disconnected at the tube, and are snaked through the gantry and a pain to get in/out. Tie kite string to the ends before you pull them out so that you have an easy way to fish them back through for re-assembly.
All the other cables can be disconnected at the board next to the cable tray.
Keep in mind that removing the tube from the gantry or disassembling the mirror enclosure will have an impact on the alignment of your unit. Nobody (other than maybe Glowforge) has come up with a proven way to realign them once they get boogered up.