I'd honestly still use chaos. Not the full set, I very rarely use the full set because its suicidal when you're too lazy to dodge things, but use 1 piece of it and 1 piece of seerus/divine veil/whatever fits the level. That gets you a positive amount of status resistance, and if you find UVs for it, then you'll be very well off. It might sound expensive, and it is, but this is a game about earning gear so you might as well earn gear. The exact setup of your gear doesn't matter too much, because after a while, you'll have more than enough gear to figure things out for yourself and experiment.

Zeddy, Seerus mask has -2 poison and -2 freeze as well.

Oh good Havens, not negative freeze and poison!

As someone who goes hybrid Sword-Gun, I find that damage is the more difficult part.

Swifty's ASI High is really all you need for both guns and swords. It also gives you the potential for UV's to shine through. Also, I find it's better to work on an individual weapon basis, instead of over-arching universal boosts. Certain weapons do better with ASI, while others want CTR. For example, CTR VH on an Argent Peacemaker would be totally useless, while on a Blitz Needle it's greatly appreciated. Conversely, ASI VH on a Blitz Needle is rather pointless, while on an Argent Peacemaker it's amazing. In the end, your equipment is not static. You CAN get more than one set of 5* armor and trinkets. For example, if you're using a Flourish and an Argent Peacemaker, you can change around your equip to give you more ASI, instead of CTR, while if you're using a Glacius and a Blitz Needle, you can sacrifice ASI for CTR. You can also get UV's on individual weapons if you want them a bit faster or to charge faster.

Damage, however, is a different story! You cannot get a universal damage boost as a UV. You have to rely on other equipment to do so. I usually use Skolver/Snarby in combination with trinkets to boost gun damage. You can also throw in a piece of Chaos Armor if you are not too concerned about defenses. I also like getting the occasional +dam med vs a favorable monster family UV on my weapons, like +dam med vs Gremlins on an Acheron. You can always fall back on UV's for ASI and CTR, but not for damage.

In general, as a Hybrid, it's better to aim for universal VH on as many things as you can, instead of MAX. VH, or even just High will give you all you need in general, while still allowing UVs on individual equipment to shine through. If you're good at not getting hit, Swifty and the Chaos Set will be your best friends ever. If not, the Barbarous Thorn Shield is great as a piercing shield for the sword damage boost. For other damage types, just go with whatever shield you want. If you don't want a shield per damage type, just go with something like the Volcanic Plate Shield or Omega Shell.

TL;DR:
- Focus on getting universal damage over ASI and CTR
- Aim for Very High on as many things as possible, instead of MAX
- Swiftstrike Buckler and Chaos Set is amazing for going hybrid
- Think about each individual weapon and it's needs instead of universally boosting them all

Hope you don't mind me slicing up your comment a bit here...

"Now come the questions. Would getting an ASI MAX (on a weapon that actually benefits from it) give it a de facto damage boost, since its firing more bullets/swinging more often in the same time period as a weapon with a damage boost?"

It's kind of hard to compare attack speed and damage boost when figuring out your overall damage. It honestly depends on the weapon, the situation, and just personal fighting style. The only time I like really high attack speed over damage is when I'm trying to get out as many full combos with my sword in order to stagger a target (or just do as much damage as possible). However, you rarely are able to just combo endlessly. Also, when you have a high enough damage boost, it'll sometimes take just one less swing to kill an enemy. In that case, the damage bonus wins out over the attack speed. I still think you want SOME attack speed, which is why Swifty is just AMAZING, but I honestly prefer damage over speed.

The exception to this is debatably the Flourish which benefits so much from attack speed and shield canceling. You can also debate alchemers, if you're doing the weapon-switching shenanigans (I prefer to charge spam with my Alchemers).

"Does getting a coordinating damage bonus (Say, bonus to constructs on a Polaris) really add enough damage on top of the critical damage you get from the weapon's own damage type to make a noticeable difference?"

I feel that it does. As mentioned before, you'll end up needing one or two less attacks to kill an enemy. The reason why I like charge spamming with Alchemers is, if your damage is high enough, you're essentially insta-gibbing with every shot if the bounces cooperate. Often, a single charged shot from my Nova Driver, with a combined total of +dam Max vs Undead, can insta-gib (or nearly insta-gib) several FSC zombies, even in a party. That's not really possible (or at least rarer/harder) when you don't have the damage bonus.

"Could you clarify what you meant about aiming for a VH bonus instead of max?"

I mean you want your armor, trinkets, and shield, to give no more than a VH bonus. As a Hybrid, you have a lot of things you want to boost, yet you can only get so many bonuses. Thus, I find it's better to get High or VH on as many things as possible, instead of getting MAX on just a few of them. You can then use UV's to get that little extra bonus to reach MAX if you really want to.

In the situation with the Blitz Needle, I'd still say get the CTR Gun trinket, just so you have an overall CTR VH on all your guns (If you include the bonus from heat lvl 10). The Blitz Needle relies a lot on its charge, thus you have the UV to bump just the Blitz up to max. This way, you rely on the UV to boost you to max, instead of using a second precious trinket slot to benefit only one weapon.

Go with Zeddy's idea but get freeze and poison uvs.