Well, remember that there are no hard and fast rules here -- you're the "man on the ground," so to speak, and you're going to have a much better view of things than some commanders back at HQ, right?
The other thing to bear in mind is that you're in a cage match, very literally -- this is precisely what I meant on the wiki. You have stuff that you need to capture, and the AI is right freaking on top of it with hard planets, core planets, or homeworlds. So that means you have an exceptional challenge and you might need to make some hard choices that would normally be avoidable on a 60 or 80 planet map.
Case in point: if there's a bunch of stuff that you need that's next to core worlds (
not homeworlds), then I don't see that you have much choice but to take that stuff. Neuter as you can, or just deal with the blowback. Your idea of spreading the alert around a bit is a good one, but core worlds will get some precedence anyhow, just FYI (not as much as a homeworld, but more than regular planets).
If you're trying to capture advanced factories or fabricators or something that you have to hold a specific planet in order to keep, I'd leave those as long as possible. It doesn't seem likely you'll be able to hold every planet that is next to a core world, so go for the ones with things like ARSes first, then abandon them when the going gets tough so that you can consolidate on the planets with the factories or fabs. Same thing thing with golems: get in, rebuild the golem, and get out unless the planet turns out to be easier than you thought.
The end result of that is that these core worlds are going to give you more of a problem at the end of the game -- that sounds unavoidable at this stage. But the other end result is that you'll actually be equipped to deal with them. But my rule of thumb would be to alert each core planet as late as you can, whatever that turns out to mean -- if there are other ARSes and things elsewhere in the galaxy, get those first.
I do very much think that you can win the game without mark IV ships, by the way -- I've certainly had to go down that road many times, after my factories got destroyed. In that case, it's sort of a tossup between the mark IIIs alone versus starships. Depends on the starship and your playstyle, and honestly the balance has shifted enough lately that I'm not sure of my answer because I haven't been in that situation in a while. Starships have gotten more useful lately, so my default of gong with the mark IIIs may not be correct anymore.
Frankly, I think that trying to neuter a bunch of core worlds is going to be a bad route to go down, but it's up to you -- whatever makes the most sense in this scenario. I'd just alert them and then deal with the backwash, occasionally clearing out the wormholes leading to your planets if they are getting a big buildup. I would strongly advocate leaving a scout on each core world to keep an eye on things, by the way. But I just figure that if you're trying to neuter so many core worlds, you're going to wind up leaving the homeworlds on alert too long because of the proximity of your fleet, and at the same time you're going to be expending a lot of your own ships and time for something that will just keep regrowing.
If you want to punch a hole through the core planets, the smartest decision might actually turn out to be to take them -- against the general advice. This, again, I've had to do many times. Sometimes a core world is just too strong for you to be able to route a bunch of guys through, and so you're going to just have to suck it up and take the whole thing out, then deal with the fully-ready homeworld on the other side.
Homeworlds don't do border aggression to my recollection, so another (rather extreme) route would be to take all of the core worlds and then consolidate with your massive throbbing economy and all those ARSes and factories and fabs, and then roll the homeworlds that way. You'll be playing a very long game, and a very risky high-AIP game, but it should be pretty exciting and definitely is winnable.
I've never used this strategy, but I know other players who have.
In the end, a lot of this comes down to playstyle. How are you at defending yourself? What is your defensive position on this map?
Do you have a good, long bottleneck between your homeworld and other planets? If you've got a strong of 3-4 planets that the AI must go through in order between any of their planets and your homeworld, then you have some latitude to stir up the AI and take some backwash. You can use a boom strategy where you try to drive your economy and your technology and your fabs, etc, sky-high while the AIP also goes abnormally high. Then crush them while using your bottleneck to provide some insurance against the AI aggression. Broadly, this is a playstyle I use a lot.
On the flip side, are you really exposed on your own homeworld? Is there only 1 or 2 planets buffering your homeworld, and/or does the AI have multiple routes to reach your homeworld? Are you particularly skilled at surgical raids and other offensive cleverness? In this case maybe you just ought to take what you need, alert the core worlds, but then strike through or past them without letting the AIP get too high or too much time go by. This will be a faster game, but riskier in its own way. And I tend to be a boomer, so it doesn't match my own style much, but many other players use this in varying extremes with excellent success (often playing higher difficulties than I do).
In the end, there's no wrong answer as long as you don't let the AI punch through to your homeworld, and so long as you don't put yourself into a stalemate situation where you have a lot of ships but are unable to punch through to the AI's homeworld. A lot of the changes from 4.0 and on have made stalemate situations vastly less common, though, so my guess is that you'll either lose or win, not get stuck, one way or the other.
Hope that helps.