I am aware of two major bug-fighting strategies. One is to use Vertigo Pulse, which is extremely effective, but relies on having a psycaster with vertigo pulse, something that is absolutely not guaranteed.
If you have a psycaster at all chances are they have something useful. Blinding pulse reduces melee hit chance, chaos skip can pull them off your people, etc. Once with some teamwork I managed to blink a psycaster in, berserk a megaspider, blink out, and watch it fight all the other bugs without ever agroing on a colonist. But yeah, in general vertigo pulse is my go-to and I buy psytrainers for it whenever I can.
I have two strategies for you:
1. Animal ganking. Train anything, even rats, to attack and send dozens to take the place of your melee fighters in your scenario. They will almost certainly die, but better them than a colonist. Once an enemy (any) is engaged in melee with someone it won't stop attacking them to chase someone, so your shooters can retreat to a safe distance once the animals arrive.
2. Shoot from afar. if you have people near the infestation when it starts, use the time before bugs arrive to have them deconstruct walls and uninstall furniture where they can to give your people a clear shot from as far away as possible. Then have your melee fighters line up right in front of your shooters. By the time the bugs get to you they will hopefully be weakened enough that the team can take them out before they do too much damage to your front line. You can also snipe them in the right circumstances, if your shooter is a good hunter they be able to stealth it.