I don't think you can save enough gas with either the V6 turbo or the LS1 to begin to pay the cost of putting one in your Cutlass.
Even if you could get the engine for free, you would be hard pressed to install it without spending enough money to buy a lot of gas.
While either the V6 turbo or the LS1 would offer considerable improvements over the stock 307, neither would provide significant (over 25%) improvement in gas milage over a well tuned 307.
Keep in mind that most of today's cars (since about 2001) achieve better performance and milage with extensive computer controls, fuel injection, and overdrive transmissions.
My 2004 GTO (w/LS1 and M6) can easily deliver 25mpg on the highway (unless I cruise 80+ mph). I get about 15 mpg in town if I keep my foot out of it.
To achieve similar results in your Cutlass you'll need a complete LS1 engine/transmission combination with all the computer/sensor/wiring.
Then factor in the cost of adapting/mounting/connecting the engine/transmission/wiring/fuel and water plumbing/new exhaust/upgrades to the suspension(this could be minimal with the LS1), and your out of pocket a pretty penny.
Don't forget what your time is worth. All the time spent scratching your head trying to figure out how to squeeze an M6 or 4L60E trans into the Cutlass tunnel might be better spent. (In fact an extra 5 hours of paid labor each week at minimum wage would buy you more gas than you are going to save)
The best advice is to fix up the 307. (if it's completely shot - try either a replacement engine like Jasper or find a deal on a crate motor). There are some good EFI systems available that use modest computer/sensor technology at reasonable prices.
Even a properly serviced two barrel carb on a 307 should delive 15 mpg in your Cutlass.
You would have to double your milage to save about ($2500/year - I spend about $50/week on gas) so it would easily be three or four years before you would break even on a swap to a V6 turbo or LS1 IF you could get it done for less than $10,000. I think the going rate on a used (100k mile) complete LS1 is around $5000, but there may be some better deals out there.