To go any faster you'd need more tankers as well. I don't forsee the tanker having commercial value short-term, so this would still increase the cost. With two tankers though, you can just have a full tanker waiting in orbit which is the fastest possibility, so there is no need of extra boosters in that case.
I agree with the premise of multiple boosters though. I wonder if a spacecraft could be made that could do both commercial earth orbit satellite missions, and serve as a tanker. This could finance a larger number of tankers. I think you'd need a 3rd stage on top of the tanker in this case to get the satellite in position. [EDIT: I mean either carry the extra fuel, or carry a 3rd stage + satellite, not both at the same time.]
Just launch tankers and fill up tanks on Tiangong and then the space craft with passengers would just have one trip to the space station to fuel up and then go to Mars.
China is going to throw away the station anyway, so might as well give it (or lease it) to SpaceX for a good price in exchange for at-cost refueling for their own ambitions.
If you're going to Mars, your vessel has a reasonable crew habitat, so you don't need Tiangong I for that.
Tiangong I is more payload than spaceship. Its tanks can only hold about 1000kg of propellant. Mr. Musk's proposed transfer stage has a propellant budget that's nearly 2,000,000kg.
Having to maintain compatibility with it may be hard (comms, docking, etc). There might be some hardware you could loot off it, but it's not worth flying a mission just for that, and launch windows mean visiting on a trip somewhere else would be prohibitively expensive.