Fanatical purifiers favor kinetic weapons and defensively they have a slight preference for shields over armor. You can check that out on the wiki page on AI personalities: https://stellaris.paradoxwikis.com/AI_personalities#Regular_empires
Design your ships to have more armor than shields and maintain a balance between kinetic and energy weapons. If one is more advanced than the other, lean into the advanced one. For example if you have Railguns and Blue Lasers you'd want to use more of the former, but if you have Railguns and X-Ray Lasers you'd want to use more of the latter.
At equal tech level energy weapons are a little better than kinetic ones because they're better at focusing fire on damaged enemy ships (they prefer shooting ships that have already lost their shields) and perhaps more importantly, starbases get bonus armor so energy weapons are better at killing them, which makes them better for pushes into enemy territory.
There is no special point to equipping corvettes with one type of weapon and destroyers with another unless you're going to keep the fleets separated and run them against different targets. But if you'll stick them into the same fleet, then just make both designs balanced between energy and kinetics. In fact, if you have destroyers you'd probably want to stop making corvettes. Even better is if you can wait until you unlock cruisers. Carrier cruisers cause a revolutionary in military technology when they first become available and if you can afford to turtle behind your starbases you'd best wait until you research them before investing more alloys into ships. If you must build ships before that, build destroyers.
Also, once you have a fleet that you think can take them on, try to be the aggressor and invite friends. It will give them someone else to shoot at. Another trick is that if the enemy is affraid to attack your starbase + fleet, you can move your fleet to a system next door and park it right next to the hyperlane entry/exit. Then if the AI falls for it and engages your starbase you can jump in with your fleet.
And speaking of starbases, you did equip them with hangars, right?
Design your ships to have more armor than shields and maintain a balance between kinetic and energy weapons. If one is more advanced than the other, lean into the advanced one. For example if you have Railguns and Blue Lasers you'd want to use more of the former, but if you have Railguns and X-Ray Lasers you'd want to use more of the latter.
At equal tech level energy weapons are a little better than kinetic ones because they're better at focusing fire on damaged enemy ships (they prefer shooting ships that have already lost their shields) and perhaps more importantly, starbases get bonus armor so energy weapons are better at killing them, which makes them better for pushes into enemy territory.
There is no special point to equipping corvettes with one type of weapon and destroyers with another unless you're going to keep the fleets separated and run them against different targets. But if you'll stick them into the same fleet, then just make both designs balanced between energy and kinetics. In fact, if you have destroyers you'd probably want to stop making corvettes. Even better is if you can wait until you unlock cruisers. Carrier cruisers cause a revolutionary in military technology when they first become available and if you can afford to turtle behind your starbases you'd best wait until you research them before investing more alloys into ships. If you must build ships before that, build destroyers.
Also, once you have a fleet that you think can take them on, try to be the aggressor and invite friends. It will give them someone else to shoot at. Another trick is that if the enemy is affraid to attack your starbase + fleet, you can move your fleet to a system next door and park it right next to the hyperlane entry/exit. Then if the AI falls for it and engages your starbase you can jump in with your fleet.
And speaking of starbases, you did equip them with hangars, right?
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