After some incredibly un-intensive testing on max stated ranged build characters I can share some probably outdated information that most people already know. But first, a general tip for all classes: After shooting a ranged attack, the Quadomme will use her ranged evasion buff and make it a waste of a turn to keep shooting. Unfortunately melee is not a great option at this point because you can be countered, although my smuggler had a much better success rate in landing attacks without being countered. If you don't have any special abilities or consumables that you could use on your turn at this point then just wait and conserve energy. After two or three turns she will reset and her buff will go away. If you keep shooting at her while the buff is up it will reset the timer, so don't shoot! Also, if the buff is up and you have stunned or flashed her, she will still dodge ranged attacks. Confirmed for Headbutt, Paralyzing Shock and flash grenade. Another small tip is to use the time when she has the buff up to destroy the webs around you to prevent any bullshit from happening (for a little while).
For Mercenaries: Concussive Shot is your best bet. I was landing about the same damage with a concussive shot from the Korg Greatbow as I was when I landed a shot with the Stormbull, and the CS appears to have a high chance of landing even when the Quadomme has her ranged evasion up. The stun is a nice bonus as well, but remember the tip above. The trick here is to alternate the use of standard ranged attacks and CS to make the most of your energy, since she will be sapping it by making you struggle out of grapples. Use Second Wind to get back energy and keep spamming it.
For Smugglers: I tried fighting once with the Hirudo Devourer and once with the Vanquisher. Hirudo kicked ass, Vanquisher sucked. The problem is her armor. A kinetic, thermal or shock weapon will do less damage, but the devourer does corrosive damage which destroys armor. So, use the Devourer. I used Flash Grenades and got off big damage with Aimed Shots (around a hundred damage or more) while the Vanquisher struggled to stay in the thirties. The grenades work ok too but only if their armor is down. I compared damage with and without armor on and the grenades do about double the damage when the Quadomme is unarmored; ~ 65-70 compared to ~ 30-35 with armor. Really, the Hirudo Devourer + flash grenades + aimed shot makes this fight easy. (After I realized how good the Devourer was I realized my other strategies were kind of pointless compared to just "use the Hirudo Devourer and don't shoot when she activates her bullshit," but I'll leave the other strategies in just in case some people don't want to drop the 60k to get this gun.)
For Tech Specialists: Literally just spam your Gravidic Disruptor. I don't know the damage value of its counterpart, but with max INT on my build I could consistently two to three shot her with around 115-140 damage attacks. It helps that it is, you know, Unresistable damage and it can't be dodged either, even with her ranged evasion up. If you get grappled, static burst is your friend, but don't overdo it or you will find your energy depleting pretty quick.
TLDR: Use the Hirudo Devourer (bought from Kattom on Uveto for 60k), use Special Abilities as needed, and wait out the evasion buff.