Just so many things seem wrong in this idea... but as I think about it, it does seem reasonable.

When I think about it, I don't see anything wrong with the idea... except that no matter what you are doing you will hear freewheel clicking.

Also clearance on the left chainstay may be an issue (the right chainstay is sometimes dimpled for chain ring clearance).

But thinking about it while typing, I see that it would possibly work... so I guess the only real issue would be a slight delay while the freewheel engages when switching force.

OK, a real significant issue here... It might also be hard to get used to the mechanism for riding the bike. While riding forward one would apply rearward pressure to slow down, but at the same time the cadence would increase. This could actually be dangerous if one is riding at near their maximum cadence.

But wait, that doesn't make sense... If the "reverse" freewheel side is at a lower gear, then that freewheel would always be engaged and attempt to force the cranks faster than the "forward" side...

Oh, I see... that doesn't make sense since the freewheels would actually work in opposite directions... therefore you would have to pedal forward, or reverse, and have a neutral if not pedalling.

I really should sit down with a paper and pencil before rambling on, since I have just proven that I have no idea what will happen without some doodling...

Summary:

I don't know which would be the problem, but I can say confidently that it won't work in a traditional fixed gear way.

That is my final answer!

(Thanks for letting me think out loud)