Having finished the game, here is my character tier list:
S Tier- Luigi, Rabbid Rosalina
A Tier- Mario, Bowser
B Tier- Rabbid Mario, Edge
C Tier- Peach, Rabbid Peach
D Tier- Rabbid Luigi
Justifications:
Luigi was by far and away my best overall damage dealer. His range is ridiculous, and if you spec him to have three shots with steely stare that can go through cover, he can nearly one shot any non-boss with each hit. His wider movement range and multiple jumps also make him surprisingly mobile, and I got through several "reach the area" maps in one or two turns just by using him wisely.
Rabbid Rosalina's stasis ability, especially if you spec to have the dash stasis ability, let's her shut down huge amounts of the map, and makes things much more manageable as you go. Her attack is also great for blasting through cover, and does really high damage, especially when you increase the crit damage/rate and stack a spark ability on it. This isn't necessarily because of her character, but I gave her the magma rain spark to mitigate her lack of multi-target attacks, and she quickly became a terror in every match.
Mario I was worried would drop off in utility pretty quickly, but his usefulness really increases as you fill out his skill tree. His being able to attack in air gives him great approach options, and combined with his stomp ability, can give him great gliding range and damage against single or multiple targets. Attacking in air also lets him move after landing, which can be of great use. But where he really becomes busted is with gaining the ability to use Hero's sight every single turn, with multiple hits, which you can further augment with conditional recharges. I would give him the invisibility spark, put him behind enemy lines, and let him go to town with Hero's sight.
I ended up really enjoying Bowser, and he is situationally very useful. His weapon can hit a cluster and does decent damage, and his landing damage attack is useful for maps with lots of weak enemies. But the fun really begins with the rabbid mecha koopas, once you add the fire ability to them and increase their numbers. Sending them out to mess up the map was hilarious every time, and more effective than not usually.
I didn't use Rabbid Mario as much as I would have liked, but his wide angle of attack and ability to attack behind cover made him a great option for maps with large amounts of enemies or ones with lots of terrain obstacles. His counterattack I found to be one of the less useful abilities.
Edge was good for maps that required a lot of dashing and movement, such as those with lots of bob-ombs and goombas, and her sword did good damage against large enemies, but I didn't find her useful enough to use her much outside of these circumstances. Her overwatch ability is disappointingly short ranged and just not as useful as some of the other utility abilities.
Peach's Barrier ability is very useful and can be amazing on maps with either heavy hitters or spammers. But her shotgun attack is terrible; it does little damage, destroys cover, and is too short range. There are far better wide range attacks with sparks, and cover is generally more useful to your characters than the enemies. Her skill tree is focused around increasing her barrier charges and effectiveness, but Peach is at best a situationally useful character.
Rabbid Peach is probably not as bad as next to last place for most people, but at least in my experience, healing just isn't that useful. You have the opportunity to heal after every fight, healing items are cheap, coins have little utility beyond healing, and I just didn't find the opportunity cost of using her worth not using another more useful party member. And because I didn't focus on healing, I was able to use the skill points on all the other characters for things other than increased healing effectiveness.
So I initially forgot Rabbid Luigi was even in this game and didn't realize I didn't have him on this list until after I posted it. So anyway, Rabbid Luigi's weapon can hit multiple enemies, which is neat, but is fairly weak even with three hits considered, and his exhaust ability is pretty underwhelming, especially compared to Rabbid Rosalina's stasis ability. Overall, just a very underwhelming character, which is funny because I remember him being pretty great in Kingdom Battle.