Overall my plan is to use my FM fleet for missions 1 - 4 & 6. I use a conqueror fleet for Missions 5 & 7 (obviously).
My FM fleet has been bolstered by the 12 days worth of Punisher build tokens. I used those to convert 2 Punishers from Draconian Uranium Base focused Arbalest ships to more tanky ships I can use in the FM. With just one well built Punisher tank I could just about do a full round of the FM without stopping to repair. Two 6 day refits left me with 2 more Punisher tanks that aren't as good as the original but are still decent. I burn through those weaker two tanks on the first four missions and then save my good tank for #6.
A Long Pinch XL doesn't hurt on some of those islands either. I don't use rockets that often these days... but why not?
Bullseye Brigade can also help a lot, but I hesitate to recommend rare or legendary crews as the key for anything, since many players can't count on having a specific one of those.
For mission #6, the Dreadnaught X can really lay down a lot of mortars and really tear you up. On my first runthrough it totally overwhelmed and destroyed my Countermeasure tank. The trick I used the next time.... I ran to the Southwest as I was engaging the other ships, and I was able to retreat before that DNX did any damage. Then I came in with Hellwraiths. Although that ship has thermal, my Hells were able to outrun the shells no problem and sink that bastard (it doesn't have much health).
For the bases, on #5 I used a Spite to spot for Vendettas. After clearing the front turrets and the ship, I ran the Vendettas in behind - far enough back that they were not running into the fire pools. As the Spite got in range of the dock, the Vendetta let off a might blast of mortars ad killed much of the center turrets. My Spite took some damage on this.. but not too bad.
On base #7... Metatech or whomever that is made a big noob mistake in their base layout:
Yes, prep that right side. After that you can prep those front four turrets as well without coming under too much fire. Take your time because they have lots of health.
Since I have a Pride built with its specials and a couple Ruin Scatterguns, I was able to blast through this base with nothing more than instant repair on any of my ships... after I went around that first bend I took it slow with the Pride to hit each turret as I went, then brought up the Vendettas again to finish it off. Without the Pride you'll be somewhat worse off, but look for something that can resist those Maxim guns.
I hope that helps you out with the Homeland campaign - you've got less than 2 days left to take some more shots at it. If you miss out on the Liberator... well I think there will be more Tier 6 Conquerors coming, so you won't be left out forever.