Man, you're getting into dangerous territory here. Gray-water systems are usually very difficult to get approved, and require constant inspections. A new sewer tap is probably out of the question as well. What you probably need is a 'chopper pump' to raise the waste up to the level of your existing sewer tap. This is not a big deal; here in Austin there's a number of areas where the houses are lower than the street (where the line is), so they've used chopper pumps for years with no trouble. You're gonna spend a lot on trenching no matter WHAT you do, so you've got to decide how bad you want this extra facility. And it sounds like you're within city limits (and codes), so you're gonna have to have a plumber do a lot of the work anyway ($$). But if you have a plumber do it, and something goes wrong, he'll fix it. If YOU screw it up, you could be in for a real (stinky) treat. Not to mention the venting requirements; VENTING was the biggest issue with my installation, not the drain. There's a whole host of code rules for drains and venting; not an area I'd want to guess about.
My house was built in 1922, and a couple years ago I started having slow drainage. A plumber came out, took the toilet off, and ran a camera down into the drain. The problem? The drain line was made of stuff called 'Orangeburg' (after the town where it was made), which is basically cardboard tube soaked in tar. After eighty years, it starts to fall apart. They replaced the whole shebang with PVC, and it's unlikely I'll ever have to think about it again. Apparently Orangeburg was used all over the country for MANY years, I think they finally quit using it when PVC came on the scene in the seventies.
I've never had any qualms about running my own supply lines, but I am VERY reluctant to mess around with drain lines; I'll hire a professional to do it. The consequences of a leaky supply line? wet face and floor, so what. The consequences of a leaky drain line are FAR more severe...
Good Luck!
c.