I have no experience in this, but will offer a few thoughts just the same.
I'd guess you could dig the hole fast enough with the backhoe. Getting rid of the dirt is going to be the real chore.
Not knowing your soil conditions probaby makes this point mute, but I'd dig about six inches to a foot deeper then you need, then fill with sand.
Instead of trying to dig the exact width for a tight fit, which will be impossible anyway, I'd dig it with a nice slope all the way around. Give yourself plenty of room to install the plumbing and electrical when it's in place.
I have no idea how you get the pool into the hole, but you might consider a long ramp if you don't have a crane or something to pick it up with.
After you get everything connected and tested, I'd put a litte water in it, maybe three to four feet. Then I'd start backfilling with sand.
Sand is the only material that will compact on it's own to the upper 90 percent range. Get good clean corse sand.
Fill around the sides a few feet at a time, then jet with water. I did this for awhile in California running a water truck on construciton sites. Basically, you just shove a pvc pipe into the sand with water running through it. Your garden hose will supply the water. Push the pipe into the sand pile and after awhile, the sand will collapse into itself and voiding the air pockets.
This will take ALLOT of water, but it's the ony way I know to be sure to get solid contact all the way around your pool.
It's also important to keep the pool full of enough water to avoid it "FLOATING" on the water your putting into the hole.
Keep adding water as the sand level rises, but not so much that the water lever is going to cause the pool to flex because theres nothing there to support it.
I don't see why you couldn't do it, but I'm the type that tries to do everything myself anyway. Sometimes it works, sometimes I get into trouble.
Good luck,
Eddie