Two guys with very sensitive shotgun mics, one on each side, could handle this, each operator covering one half of the set. Not perfect, but better than a fixed camera-mounted shotgun.
This would be a good scenario for boundary mics, which can cover 2-3 people each with good results, but you need a place to lay them and your set doesn’t have that. if you had little tiny cocktail tables interspersed between the folks, one table for each three people, the mics could rest on those. They lay flat and are not very visible.
I think how I’d approach this, if I couldn’t mic each person, is I’d give the host their own lav mic, then hang some omni- directional mics from overhead, just out of shot, one mic for each close “cluster” of 4 individuals, then you could keep each of those mics on a separate, recorded track, or feed them to a live person to mix in real time, or to an auto-mixer.
The auto-mixer is something that’s evolved from phone conferencing technology: it listens to something like 12 or more mic inputs simultaneously, and when it hears one of those start to have more signal, because it’s nearest the one talking, it ducks the other mics and brings up the talker’s mic in a space of a millisecond or two, then drops the talker’s mic level once it no longer detects a steady level of speech. We use one called “gator”. It’s not perfect, but it does the job.
Since you have two or more rows of people talking, it could be possible, instead of hanging mics from above, to hanging omni lavs on the backs of some of the people in the front row, to pick up 2-3 people immediately behind that person.
Part of your calculation of expense has to consider the savings in getting the live audio more or less right in real time, live, versus the longer time it takes to mix in post. If your post time is essentially free, you can afford to do it there, sure. But it can take three hours to mix one single hour, perhaps longer, if the conversations are rapid and there’s problems to deal with here and there. It’s not always immediately apparent, which path ends up really being cheaper.
As to retroactively improving the existing shows, and helping fill in gaps in the coverage of your future shows, I’d strongly suggest investing in the audio app/ plug-in called Izotope. A little pricy, but the results are very worth it. It can take out the hollowness of off-mic sources and restore problematic tracks.