My other option might be to glue a rougher grit paper to a large flat board and use this combined with a lot of elbow grease to smooth it out.
If you use the right abrasive and the right strategy this basic idea could work well for you, without it being tons of effort (as hand sanding usually is).
You must ensure the abrasive is taut, so it doesn't ruck up as you push back and forth, so it may be necessary to glue it to the substrate.
Also, instead of making one or more large sanding boards you might want to make large sanding blocks instead. Easier to hold and manipulate sanding blocks than the whole board.
Don't use cheap paper here, it's a false economy. It might be worth investing in some 3M Cubitron for this!
Start coarse
In general with something like this it's advisable to begin with a coarser grit than you're probably thinking, possibly much coarser. But a very good modern abrasive like Cubitron changes the picture.
Start by sanding on the diagonals
Wood is much easier to sand (and plane incidentally) across the grain rather than along it. Not sure if you've alternated the grain direction on your white and black squares as is often done, but if you have the ideal compromise is to sand at 45°.
Grit progression
Suggested grit progression for the hand sanding is 60, 100, 150, 240 if using conventional abrasives. If you use Cubitron or something else high-tech you could probably get away with starting at 100.
Bring out your ROS with finer papers to finish off
You might consider going to 320 or a little above here, although that's generally finer than you need to sand wood.
Alternatively, scrape after board has been made flat
Good results can be had with a well-prepared card scraper, or with the sharp edge of a utility-knife blade or single-edged razor if you don't currently own any card scrapers.
