Fitting in a tour before 5PM flight

First off, weather may be a factor in mid-November, I would be aware of that.

Second, it's ~45 minutes from downtown to the airport.

If you can find a tour that can pick you up/drop you off at the airport, that could
help, you could drop your bags off early in the morning.

Otherwise, I'm guessing you have till around 1pm until you need to leave downtown
to head to the airport.

There are no regular tours that will work for that, but you could always contact tour operators and check on a private tour. As mentioned, you will need to leave Reykjavik no later than 1 for a 5 pm flight. You could also check and see if they would take you to the airport from the tour, which might save you a bit of time.

Most group tours that time of year don't get going until 9 or 10, and Golden Circle tours are in the 5-6 hour range at a minimum, usually much longer. As mentioned a private tour may be your best bet. Here's just one example, from a company I used for a food tour (and yes, private tours are spendy): https://yourfriendinreykjavik.com/private-sightseeing-experiences/

CL--We also used them for the food tour twice. They are a great company!

Maybe we should rent a car then? We can see some sights and then go directly to the airport?

You could, but mid November the weather could be atrocious. We have had 4 winter trips and have extensive winter driving experience, and I know not to count on driving there in the winter until the day of, after checking the forecasts.

The Blue Lagoon is fairly close to the airport, as is the area where the most
recent volcanic eruptions have been taking place. Those could be options
for seeing something on the way to Keflavik.

I do think that a golden circle visit would be too short to appreciate what you're
seeing (or see all the sights effectively), and you'd be looking at your watch all the time.