A 2 foot trench 650 long is not a very big project. I've done 1,200 feet with a walk behind trencher in red clay in a day including installing the pipe and backfilling. I've dug 4 foot trenches 500 feet long in half a day with my backhoe. It's a Ford 555E.
I would consider your 650 foot trench a half day project on my backhoe. The biggest issue in time trenching with a backhoe is moving the backhoe into possition. For me, I just lift the bucket and outriggers, then push with the hoestick. It takes a minute, but when you repeat it every dozen feet or so, it adds up.
What you buy and how much power you have will factor into the time. Red clay is pretty tough digging, but conditions vary around the country.
Have you asked your utilities if you are allowed to dig the trench yourself? It varies from place to place what you can and can't do depending on policy, liability and the mood of the guy in charge on the day you ask.
I can dig all my water lines but not my electrical ones before a meter. After the meter, I can do what I want.
Septic lines are part of the system and the contractors all dig those. Doesn't matter how many hours I have or the level of my ability, they just wont allow me to dig those trenches.
I still haven't read a good argument for spending the extra money for a backhoe in your situation yet. If you've got the cash and it's something you really want, then it doesn't matter.
Like everyone who has a backhoe has said, it's nice to have and you will find things to use it on, but is it something you need right now?
If it's part of the budget, and you're sacrificing on the tractor you want in order to add the backhoe, I would definately go for the better tractor. Wether it's more HP, some extra options or larger mower, I'd consider upgrading in this area.