Ok, first, you need to tighten up your system defenses. I played through 4 waves, and while I defended successfully, the systems were in no position to protect themselves. I need to swap turrets around, force fields, and such. There is no reason not to get one of your two Modular Fortresses down next to your Advanced Factory. Speaking of that system, you've got Sniper Turrets under the CS force field. Delete them and put them at system edge. Also, you've still got a Mark II Force Field left, use it on the Adv. Factory. Also, as soon as that Missile Frigate wave hits, order all your Sniper Turrets to target the Raid Starships. This is generally a good idea for Sniper Turrets anyway, but it is critical in systems with something to protect. Sniper Turrets wreck Raid Starships.
As for the homeworld, I'm sure there are a few good methods, but this one will work for you fairly easily, although it is micro heavy. If you've never done stealth ops, this is how it works:
Step 0) You need Scout Starships to negate the railguns on the Teuthida. You've got Scout Starships wasting time on picket duty so delete them (leave the one on the other homeworld). This gives you 4 Mark I Scout Starships. You could use Mark IIs since you unlocked them, but they are more expensive to replace. Your call. Anyway, build a Starship constructor at the system you own in the top right. Crank out 2 Cloaker Starships and 4 Scout Starships, and send them all to the Core World you are staging on. Also, move that Warhead way out to the system edge so it doesn't get destroyed easily.
Step 1) Kill that Tachyon Guardian. Use a full cap of Fighters I (to absorb damage), Missile Frigates I, and Heavy Bomber Starships (to do the killing). Power them all down, and bring over your 4 Scout Starships. Try and use Mark I Scout Starships since you are likely to lose them all. Pile them all up on the worm hole and get ready. Send one of the 4 scouts through, and right as it touches the worm hole, pause and order the rest of the group through. Then switch screens and unpause. You want most of your control group through. As soon as you see the ship count under Control Group 1 (or whatever # you picked) hit about 175, pause (96+96+4=196). Order them to attack the Tachyon Guardian and shift+rt.click the worm hole so they flee right after. Then power them up and unpause. Once it blows order your Scout Starships to retreat. I was able to get some of the fleet ships out and all four of the Heavy Bomber Starships survived.
Step 2) Build 4 Transports at your staging world and send them over to the core world with your Plasma Siege Starships loaded up. Once they get there, pack two of the transports with the following powered down ships: all 3 Raid Starships you have, all the Core Siege Engines, all the Plasma Siege Starships. This, plus 4 Scout Starships and 2 Cloaker Starships are about to do some heavy lifting.
Step 3) Send your Mark II Scout STarship through to sit on the worm hole. This will reveal all the mines, and protect anything you send through from sniper fire. Next, send a Cloaker Starship through. Be certain it doesn't hit a mine, so grab control of it immediately and move it somewhere safe, but still close to the worm hole. You may need to send in a cap of Fighters to set off the mines if they are too much of a problem. The AI won't rebuild them quickly (if ever). Now send the 2 transports, 4 Scout Starships and the other Cloaker Starship. Keep them all out of the mines and use G to group-move everything except the Mark II Scout Starship. Fly a C route to the left around and up and to the left of the Teuthida, right at the edge of the system. Spread your Scout Starships out to provide good counter-sniper coverage. Unload your ships and make sure they all get cloaking. You don't want to do this too close because sometimes it takes a second and something will get a shot off. Put your Raid and Plasma Siege Starships away in the Transports.
Step 4) Group move the mass into position on that Orbital Mass Driver, this includes the Transports. With the Siege Engines powered down, make sure they are in range of the Orbital Mass Driver. Order them to attack it. Nothing should happen. Move your Transports around so they are really close to the Siege Engines. Make sure you've got full Scout Starship coverage. Select the Siege Engines and power them up. Nothing will happen for 10 seconds. Then they will fire, lose cloaking, and the AI will get pissed. Make sure you let all 15 Siege Engines fire. Then pause, power them down, and order them back into the Transports. The Orbital Mass Driver will die, and you can retreat to system edge.
Step 5) From system edge, you are going to need to wait out the re-cloak delay with your Siege Engines. So unload them and watch as everything charges at you. At the last moment, reload them and let everything retreat. After about 30 seconds, do this again. You should have enough time to get them cloaked on the second try. Let everything settle.
Step 6) Optionally, go back and repeat steps 4 & 5 to take out the Warhead Interceptor, just in case you later want to use some warheads.
Step 7) Bring your stealth ops squad down to the Ion Cannon. This time use your Raid Starships instead. They will kill it nearly instantly, so just order them to attack and retreat back into the Transports. The Warhead Interceptor was reducing Raid Starship damage by 75%, that's why they wouldn't work very well last time. Repeat step #5 with your Raid Starships to get them cloaked. Make sure to power them down.
Step 8) You are now going to attack the Teuthida just like the Orbital Mass Driver. The difference here is it has Radar Dampening so you need to get a lot closer. Also, it is going to take 6 rounds to kill, since you do about 19% per volley depending on if you can get your Plasma Siege to fire on time. Use your Siege Engines powered down to find the range you need by clicking on the Teuthida and waiting for them to stop. Make sure you keep your Cloaker Starships with them. Remember, it is dangerous to unload in range, so unload everything out of range and "walk" them in under the Cloakers.
Step 9) With the Siege Engines in range, but powered down, take a shot just like you did against the OMD. Retreat them into Transports. Now have your Plasma Siege Starships take a shot once things cool down, and retreat them. Then do your re-cloaking dance at system edge. Repeat until dead.
Ta-da, Teuthida dead and no super weapons used.
If you want to make your life easier, I recommend unlocking a Mobile Repair Station. Leave one on the Core World as it will help you keep everything health there much easier. But bring another with you onto the homeworld. It is cloaked and can repair while cloaked, so you can use it to repair all your stealth ops units during the re-cloaking dance. This means you don't need to worry about attrition cutting your fun short. Unlocking Mark II Cloaker Starships might be helpful too, especially if you want to increase the size of your stealth ops team. Remember to bring only Reclamation immune units.
I've attached my save of doing all of the above, up to taking the first shot on the Teuthida. You'll notice I've got a bunch of enemies dragged up toward my preferred re-cloak dance spot. I'd need to shift left or right a decent bit to do my dance again. This save is an example of a slightly messed up shot because I got my Siege Engines out of sync, but I was rushed for time. Also, a wave knocked out Murdoch, mainly because I didn't have time in real life to deal with it and I wanted to give you a save with a solid hit on the Teuthida.
Any questions, let me know.
