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The only reason I can see it not working is that chain length is going to be different between the rings, so you'll have no chain tension on one ring unless you move the wheel in the dropouts.
A chain tensioner doesn't work with fixed, only single speed.
dglshrn-
^ this.
what you want is impossible to do well. Chain tension needs all parts to stay in their place. Chain tensioners don't work on fixed gears because they cannot handle the forces on the chain being applied 'in reverse' i.e. when skidding/slowing down.
As you probably know, a slack chain can jump off the cog/chainring. When this happens it easily gets caught between the cog and the chain stay, resulting in a crash/very sudden halt.
Two-speed hub might be the way to go :)
edit: some more thoughts on this here. You could go for the 'ghost ring' option but when you're also gonna shift the chain and thus change the chainline and have that awkward moment when its between rings, I just wouldn't...
Smolders
This seems so obvious that it seems strange no-one has already done it.
I am looking to fit a twin chainring to my fixie with a front cage shifter and a friction changer (none of this indexing - keep things basic, as is the fixie ethos).
I can't see that the lack of a freewheel will have any bearing on front shifting and if the C/L is set for dead center between the rings, there should only be a few mm variance to line on either ring.
I can see that there will be slight extra wear and tear on the chain and cogs - but apart from this I can't see any reason this setup wouldn't work..
Before I go out and spend £100-or so on components, has anyone tried it already - or any heads-up as to whether it could work?