FInally beat Challenge 6 with a time of 17:20 and got the Vanquish Platinum. It was really annoying, I rage quit for quite a while and most days I would start it up and just be too annoyed to play. It probably took 6-10 days of serious practice to perfect it. I thought it would be more satisfying, but it lost its appeal when I got to round 5 for the 40th time just to die, lol. Here's a play-by-play of the steps I did, hopefully it helps anyone.
My strategy is from a mixture of a few videos and some of my own techniques (and of course tons of practice :P)
Round 1: Boost-slide to weapon rack, switch AR for LFE Gun, use LFE Gun to wipe out spawns, wait until spawn light gets thick and immediately as you see their health bars. Use LFE gun to clear up any remaing Grogies. Switch to shotgun, run around facing backwards and mash square as he runs towards you, hit with shotgun and use boost+kick, he should suffer enough damage to fall to the ground, hit him more and boost-kick if you have enough energy, repeat as needed. At round end, pick up LFE gun ammo.
Round 2: Get behind Statue and throw an EMP at each group of bots before they spawn, then shoot at the Bogey, being sure to go into ARS to take out his beacon things when he says "Aim carefully." (This round will take practice memorizing what to do when the Bogey says certain things) Clean ip any enemies with your HMG, switch shotgun for AR before round ends.
Round 3: Hide behind destroyed craft and remove all Romanov's heads. Kill remaining Grogies and pick up ammo for AR and HMG.
Round 4: Wipe out rolling bots with LFE Gun blast, immediately glide over to the right and take out the turret bot coming up from the cliff then take cover behind the barricade in corner. Look at radar and wait for Blue Grogies to form a line or cluster then throw and EMP, hit Blue Grogies with one or two LFE shots then glide over to starting position (near stairs). Around this time a turret bot should be coming from the left, take him out. Now glide and run while kkilling any Blue Grogies remaining. Use HMG to kill the BIAs and get ammo as needed. It'll take a while but you'll slowly whittle them down. At the end of round, restock EMPs, and switch out HMG for Rocket.
Round 5: Start with Rolling bot spawn on right and wipe them out, now (this is very important), throw an EMP as you're walking towards the other rolling bots (try to aim it at the Bogeys, but it doesn't really matter), now, ignore the other bots and immediately boost towards the statue (this is also very important, DO NOT go to the stairs). Roll around to get health and energy back and take out the rolling bots as they come with LFE. Now, for the Bogeys. You're going to want to roll and boost around the whole time, making sure to constantly be moving (especially after they throw the beacons that grain your energy). Use Rockets to take out Blue Bogey as he gears up for attacks (make sure full lock-on is achieved), don't worry if you miss but try not to and be very conservative of your rockets. When you run out, pick up green upgrade cube and repeat. Right before you kill Blue Bogey, remember his position so you can pick up his rockets he drops. Now repeat these same steps on Red Bogey, staying around near the stairs and always stay opposite of Red Bogey. Whenever you have free time, switch LFE gun for Lock-on laser. Use remaning rockets then switch to Lock on Laser. Roll around a lot and use ARS mode to fire Lock on Laser. Slowly whittle away his health and be very careful.
Hope this helps and good luck. :)