Well, you do have the DLC weapons, so you should be more than willing to abuse them.
I recommend starting with ChillMan. He takes 2 damage from Screw Crusher, you can pick up a lot of screws, and his stage on hard mode doesn't get hairy until the final stretch.
During that final stretch, take it easy and use ballade cracker to kill the jumpers and snowmen, and if you take your time, the bird missiles won't catch you off guard. At the end of this level is a huge jumper (which will probably destroy your ice platforms) immediately followed by a charging siren thing. If you aren't ready to hit that siren, it'll knock you into the pit. Be ready with the BC, which will damage through the shield and push it back.
After that, take on Nitro stage, which you can mow through easily enough with just Chill Spike and BC. Remember to use spikes to kill the trucks and crackers on everything else.
From there, I go to Pump stage, which can feel very difficult the first time through, but with the DLC weapons, it's easy to charge through, terminator style. Remember not to hastily leap over pits, as some have parabombs. BC will kill shrimp instantly and won't let them counterattack. Met sludge lasts for ages on hard mode, so don't get hit by it.
From there I do Blade, as Water Shield rapes the miniboss and spikes/crackers handle everything else. Don't use Mirror Buster during the seesaw segments, you'll still get hit. Just rapid fire.
With all this, you can take Solar with ease. There's one very misleadingly tough segment near the end of the level- you need to climb a ladder with two fireball generators and a pipi (bird) flying back and forth. As soon as you enter, just keep climbing until you're between the generators on your left, and once the fireballs are out of the way, keep going. All there is to it. People mistakenly are patient, and then they never get through without a ton of damage.