The main purpose of the vent is to prevent the drain flow from siphoning the trap dry and letting sewer gas into the house. Since it sounds like a real vent isn't practical, go for a Studor vent on the drain line next to the sink. The vent must be at least 2 pipe diameters downstream of the trap. Your reasoning about using the stack as a vent is 100% correct--this would be a code violation because of the fixtures above.
Once the fixture is vented, there is no limit on the distance the drain line can run. You want at least 1/8" per foot of slope. It can tie into the stack anywhere. So you probably don't need to move the tub. I am guessing it would be easier to come in above the cleanout so you don't have to screw with moving the cleanout. Cut out a foot or so of stack and put in a 4" x 4" x 2" wye. I assume it is a cast iron stack so your options are either to stub 4" PVC out each end of the wye, or pick up a no-hub cast iron wye at the local plumbing supply--either way you tie to the existing stack with shielded no-hub couplings. Make sure you support the stack when you cut it--cast iron is heavy. Good luck!