My latest and greatest strategy so far, which avoids the artillery bug and the "i didnt loose so many fighters wtfomgbbq" bug completely (consistent victory with less than 10 spaceframes lost).
[NOOB Friendly!]1a. As soon as the mission starts, Target the
Tankers (F8);
Slave Artillery (1),
Assign Target (2) (do this for all the four tankers [just keep pressing F8 till you target all tankers]
1b. Select the
Minelayers (F11) do the same as 1a (Assign targets to the artillery)
1c. Select the two
Deimos Corvettes (F10) and assign them to the artillery (same as above) (
DO NOT assign Neptune HQ though)
--> Doing this sets up a que for the artillery right at the start of the mission (they target
Tankers->Minelayers->Two Corvettes), so this way you avoid the artillery bug altogether...and you can completely forget about the Artillery; as they pursue their "kill list".

2. Select the
Serenity through the Escort list, Tell your wing to protect it
(C-3-5).
3. Select the first
Mjolnir (F6).
3b. Target its sole beam cannon
(press K).
3c.
Fire 4 grimmlers at it when in range (about 4,200 meters...don't get too close)
3d.
Use your ammo pack. (you should be back to 4 grimmlers now)
4. Target the second
AWACs (Press F9 twice) and select its
radar dome subsystem (Press S)4a.
Fire 2 grimmlers when in range (dont get too close!)
5. By now the
Serenity should have warped clear...Order your wing to kill the second
Mjolnir (F6)5b. When they are done, select the
Radar Dome of the other awacs, and tell them to kill the subsystem
(C-3-4)6. Assign 1 fighter from your wing to kill 1
Aurora each.
(F9)(C-1-2-1)(F9)(C-1-3-1)(F9)(C-1-4-1).
By now, Serenity=saved, all the tankers should be toast, probably about 1 minelayer left (or none), awacs and aurora targeting fried.
-> All the hard stuff is done now...you're almost there!

7. Now go ahead and
scan the
Comms, Nav, and Sensors of
Neptune HQ.
7b. By now if your fighters finished killing the Auroras...give them free reign to engage at will (or
disarm the
Carthage).
-> By now you should probably have around 480 seconds until the Carthage leaves. Take a deep breath...you earned it!
Now the fun part!
8. Get to a nice cozy space around 2~3 clicks, below the
Engine subsystem of the
Carthage.
8a.
Wait for the
Artillery to finish off the two
Corvettes (their point defenses will shred your incoming gunships, etc)
-> This should happen in around a minute or so.
9. After the
Two corvettes are toast,
CALL IN DA FORCES!!!!1!1!11!1! 
9a. Call in
EVERYTHING you can find on the reinforcements list (just C-4-1 till you run out of options)...
9b. Once they have arrived...tell all of them to
disable the
Carthage (C-3-2) or (C-3-4) (<- this while selecting the Carthage's engines)9c. Watch as the
Carthage's engines drop from 100% to 0% before you can yell CARTHAAAAAGE!
10.
Free the artillery (1)...They should commence their attack on the Carthage now.
10a. Tell all your wings to
attack it
(C-3-1).
11. (THIS IS A MOST!) Accept the Carthage surrender (because you are a good person)

-> Since the strike force is in-mission when you depart, you don't enounter that other bug aswell!
So, there you have it! The greatest strategy for HFH evah!
