My spacecraft was built backwards; gimbal always lies

So you're correct in using Jool for a gravitational slingshot into a high inclination, and you're also correct in assuming that you can only get about 40° from an encounter prior to switching the spheres of influence. Trying to set up burns prior to switching SoI is an exercise in futility. What you do want to do is set up your encounter so that you're coming in close to over a pole. 200m/s early (about a month out from switching SoI) can save a lot down the road.

Getting your trans-Joolian burn correct is actually one of the biggest factors for saving fuel. Not only do you minimize delta-V on the transfer burn, but you'll also minimize delta-V on the polar injection burn. You want to start your transfer about 65° after crossing the terminator into night when Jool is leading Kerbin by about 95°. When it comes to the phase angle, burning too early is really bad, as delta-V costs go up exponentially with the amount of time before optimal. Burning late isn't so bad, as costs go up much slower (quadratically maybe? Don't know for sure). This is a utility for determining optimal burn patterns.

So you've done your transfer burn, and you've done your final correction burn before switching SoI, and you're just waiting those last few seconds before that Jool orbit line changes from yellow to green. When that happens, you'll want to immediately plop down a maneuver node not too far in front of you. You're basically going to be doing a bunch of min-maxing to get your inclination into the high eighties. First, set your incoming inclination for best result inclination. Next, add about 500m/s of prograde velocity. You're doing this because you essentially have to kill something like 4000m/s of horizontal velocity and turn that into vertical. Next, reduce your periapse using the ground vector until optimal. Rinse and repeat, adding the most velocity in the prograde direction. Eventually, you'll get to a point where the inclination of the resulting orbit is correct, but the periapse will be in the sun, or the apoapse will be way out of the system. You're three options are to try and fix it now, try to fix it at Jool periapse, or (easily) fix it once you achieve polar orbit.

It's a tough operation, but with patience and experimentation, you should be able to get it every time.

Here is the whole procedure you need to do, right from the beginning. I'm writing all this just to avoid any confusion between your initial intersects, and fine tuning your approach.

First thing here is to make sure your inclinations are as close as possible to eachother.

If you are leading the ship/object you want to dock to, burn prograde to increase your AP/PE. Your goal here is to make your orbit larger than the other ship's orbit. However, ensure that at some point, the 2 orbits intersect each other, and not at sharp angles. Try to make them as close to parallel as possible where they meet.

If you are trailing the ship, burn retrograde in order to make your orbit smaller. Again ensuring the orbits intersect at some point.

Don't go crazy with the difference in the two orbits, time and experimentation will give you more experience on getting this right. You can also re-adjust your orbits now and then by burning either prograde or retrograde at the point where the 2 orbits intersect, and seeing how it affects your intercepts.

To give you an idea how this works; a ship with a smaller orbit will make one orbit faster than a ship with a larger orbit. So a smaller orbit is "faster". So yes you were correct in your statement.

Then it's just a matter of time warping until you intersect. Make sure to right-click on the other ship/object in the map view and set it as the target. This will then show you orange and red arrows that indicate where, when, and how close you will intercept your target.

When you see the intercept arrows, you can stop the time warp and try adding a maneuver node at different spots in your orbit to try different things that may make your intercept closer. Check your distance by hovering your mouse over the orange or red arrow, make a small adjustment to the node and check to see the difference. Again, more experimenting.

Once you are within 0-10 km's (under 10 is my personal preference but it depends on your ship's abilities), you then want to click on the speedometer above the navball until it says "target" and a speed under it. This is your speed relative to your target, so if it shows 0 (zero), then you are moving at the exact same speed as your target.

Now you will point directly towards your target (purple on navball) and do a burn to get closer and raise that target speed. At this point you will see your distance from the target dropping. Now, you're moving towards your target, but your orbits are different so you will likely not hit it directly unless you were really close to begin with.

Turn your ship now to point at the yellow retrograde. Since you are in "target mode", the retrograde is adjusted in reference to your target and its orbit.

As you start getting close, you may find that you are passing beside it. This is normal. Still pointing at the yellow retrograde, you want to now burn and get your speed back down to zero. Then, point back to your target (purple) and repeat this until you are within 1KM (1000 meters) or less. Be aware that you don't want to get too close to your target when burning with a rocket because rockets can push your target away from you or send it into a spin.

Once you're within 1KM and you've brought your speed down to zero, you can now switch to using RCS only for the rest of the approach. RCS thrusters will not affect the other ship like rockets do so you don't have to worry about using them when you're close to it.

The most important part of all of this is to have patience and don't rush. If you pick up to much speed when trying to get closer, you could screw it up by overshooting your target. And it's normal to have to do several slower approaches to get closer and closer. The more you do this, the better you'll get at it.

Good luck!