Having rebuilt dozens and dozens of C4 I have NEVER seen a plug like that in the TC- there should be a regular threaded 'plug' that you remove. What you are showing looks like the plastic plug you will typically find in the pan of a C4 that had never been rebuild-it was inserted in the dipstick tube hole in the transmission casting when built and then when they inserted the dipstick in the trans on the assembly line it just pushed into the pan and remained there without issue. What you are showing looks like one of those plugs. Rotate the flywheel and look for an actual plug if there is none I would guess someone drove the plug I described into the hole possibly because they damaged the threads on the TC. The location you are showing is typically where the plug is located. I would remove that plug and collect the fluid and examine it closely if you see lots of brass colored 'dust' in the fluid-you might want to consider replacing the TC with a rebuilt one. Brass usually indicates lots of movement in the internal fins (they are brazed in place). Easier to replace it now vs later.
And yes it is less messy to drain it before moving it. You may also want to put a dummy output shaft in the tailstock to prevent any from spilling out the back of the trans.