I'm really happy with BG transmission fluid. A local indy shop I like offers BG transmission products, so a couple years ago I read up on their stuff. The reviews I read online were good, and they were even brave enough at the time to offer a million mile warranty on your transmission (obviously with extensive restrictions, limitations, and requirements). I decided I'd try stepping outside of the Motorcraft box since the indy shop was convenient, had good prices, and did good work. I've had my Ranger's transmission flushed with BG full synthetic twice now. It shifts fine with 112K miles. The shifting has never changed as a result of a flush, so that's a good thing. The current fill of BG synthetic still looks great with 30K on it, but I'm getting it flushed again in the next month or so anyway. Fords are a bit picky about transmission fluid and my Ranger doesn't even seem to notice that it doesn't have Mercon V in it.
Personally, I prefer changing as much fluid as I can at once, and I think it is highly unlikely that you will hurt your car doing that. However, a drain and fill routine is easy to do on your car, so it is a perfectly good option as well. If I remember right, the dipstick/fill hole for the transmission is right on top of it, and it has a drain plug. No mile long dipstick tube and messy pan to drop like on my truck. It probably holds less fluid too, so it would take fewer drain & fills to get mostly newer fluid in it. A wrench for the drain plug, ATF, funnel, catch pan, crush washer for the drain plug, and some paper towels should be all the supplies you need. Maybe some Rhino ramps if you don't have a good curb. That generation of Civic is parking lot repair friendly.