Is it a tall quarter barrel or a pony keg? If it's a full height, it's a bad proportionality for most gas burners. Also, the tighter curve of the smaller diameter would make it harder to get the weldless fittings to seal.
If you don't have a choice, I'm sure with enough effort it could work, any port in a storm as they say. If you know someone returning a full keg from the same brewery, you might be able to have them pull a fast one and return the 7.75 in its place. I know my local brewery doesn't keep track of the sizes for deposit purposes. You'll be much happier with a 15.5. You'd be dealing with a stolen keg, so follow your own moral compass, but you'd hardly be alone in that respect.
Most breweries wind up with unserviceable damaged kegs from normal business. A friend got two that the coupler flange was smashed, which was perfect because that's the part you cut off to make a keggle. You might be able to get one of those for way less than the cost of a deposit.