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.
A few things to note on asteroids in KSP / ARM:
The orbit is determined when the asteroid is created. But any given asteroid may not persist, or de-spawn, even in a couple days, if you don't track it.
The act of tracking it is what makes the asteroid persist for any dependable time. As soon as you stop tracking it, it may de-spawn at any time.
Un-tracked asteroids have no real effect on the game. You won't notice them impact Kerbin, or crash into one of your space stations, or anything.
Tracked asteroids have very little impact on the game. When they crash into Kerbin... nothing really happens. In fact, they are likely to be removed when the hit the atmosphere, just like any vessel or rocket debris that is not being flown or within a certain distance of something being flown.
Q: So, my question is, have I basically screwed myself by starting to track this asteroid, or would it have always ended up on a collision course, I just would have not known about it?
A: It would always have been on a collusion course, and you just wouldn't have known - although it probably would have de-spawned before ever reaching Kerbin. You aren't screwed, and have a couple options:
Keep tracking it, and watch it crash into Kerbin (for as long as it exists). It won't ruin your game, or wipe out the Kerbal race or anything.
Stop tracking it - it will most likely de-spawn before reaching Kerbin. And even if it doesn't the chances of you ever interacting with any random asteroid is infinitesimal.
Either way, you can go about your Career Mode and deal with these asteroids only when you are ready to mess with them. (Or ignore them completely, if you want).
Best Answer
If you really want a solar polar orbit, I'd probably start like you suggest, with a flyby over Jool's poles to change the inclination. You won't be able to do a full 90° inclination change just with a passive fly-by, but a burn at Jool periapsis (to maximize the Oberth effect) ought to be able to finish the job, as I've just suggested in an answer to the earlier question you linked to.
That said, if your actual goal is to build a communications network that will cover the entire solar system, there are other ways to do it that don't involve going far out of the ecliptic. For example, you could start by parking two high-power commsats in the same orbit around the Sun as Kerbin, but 120° ahead of and behind it. If you don't mind waiting a while, you can do this with very little delta-v; basically just enough to escape Kerbin's SOI, plus a tiny bit more to recircularize orbits once the satellites have drifted far enough from Kerbin.
Such a network will give you near-perfect coverage of Moho and Eve, and will also mostly cover the day sides of the outer planets, even when Kerbin itself happens to be behind the Sun. The triangle formed by the two commsats and Kerbin should be wide enough in the sky that eclipses should mostly be a non-issue, since even Ike isn't big enough when seen from Duna's surface to eclipse all three corners of the network at once. I suppose you might get occasional comms blackouts on Laythe when it goes behind Jool, though.
If you want to also cover the night sides of the outer planets, you'll have a bit more work to do. One option would be to send a constellation of three or four commsats into an orbit way out beyond Eeloo (again putting them in orbits about equally space around the Sun, in the corners of a triangle or a square, and with orbital periods matched as closely as you can). This will take considerably more delta-v than just drifting around Kerbin's orbit (and you'll need more antennas on the satellites, since they'll be further away), but probably still less than going into a polar orbit around the Sun.
The other option, of course, is to just deploy separate clusters of commsats around each planet. Honestly, this might be the simplest way to do it, especially if you can piggyback the satellites on another mission to the same planet.
(Also, by spamming enough max-level relay antennas on a single craft, it's actually not hard at all to make a commsat with more antenna power than the maxed-out DSN has. Launching one into, say, high polar Kerbin orbit way out beyond Minmus will improve your CommNet strength and also somewhat reduce the risk of random eclipses cutting off your comms at just the wrong moment.)