I have to run my guild alt through CQ, which is slightly above the OP's account in GP, but bad on recent toons.
The alt got red crate no problem, and it had very few new pirates at relic (I think just SM33, Quig, and Ithano, and only SM33 was 7-stars or better than r2). It's important to learn a few tricks for taking down squads before they can kill your weaker toons. Maul starts with Frenzy, and certain toons start with bonus turns or bonus TM. The obvious one is Wat Tambor. Use Maul lead, 2 of your best Mandalorians, the toon you need to survive, and Wat Tambor. On the first turn Wat gives Attacker-Tech to Maul, then Maul and buddies rip through the enemy without giving them a chance to do anything. This works particularly well with the Stacking Fervor disks that grant +Off% for every out of turn attack. Meanwhile, with Zealous Ambition, using Dash or Admiral Raddus as your lead allows you to one-shot entire enemy squads. Dash is particularly good since those who survive are dazed. This also works with GL Leia + Captain Rex. And, of course, if you can't trash the enemy quickly enough, Jedi Master Luke + 2 Jedi + Wat (tank tech on JML) + the toon you want to survive will win a longer battle. Jeez, I'm even forgetting SLKR + Hux + KRU + any two toons (so you can do Silvo + Vane). And there are a number of tanks with pre-taunts.
Besides just going full offence or using tanks with pre-taunt, there are other things you can do to help yourself as well. I know it seems outrageous, but if getting Conquest toons in only 3 months matters to you, you can also spend an Omicron on Padawan Obi-Wan. From learning how to best exploit
Even with all these options, I also encourage you to remember that there's no problem in getting gold crate instead. Especially on the first CQ of a cycle. You can get the toon in 4 to 6 conquests instead of 3 and using very little effort compared to what's required for red crate.
There's no reason you can't red crate, but if you don't, it's far from the end of the world.