The betta's fins are gonna look like targets to the crayfish. It's not a guarantee that the crayfish will attack the betta, but it's a bigger risk than I'd be willing to take.
How big can a crayfish get? and why would it target the betta's fins?
I'm not sure. There are a couple of types of blue crayfish. Some only get to be four inches or so, there's another that grows to be almost a foot.
As far as the fin remark, it's actually that the whole fish will be a target, but the draping fins will be easy to snag. I would guess that, at the very least, you'd end up with a betta with ragged fins.
Blue crayfish can get to be over a foot long, depending on the species.
I wouldn't put them with a betta. Honestly, the bettas are too slow and will, eventually, become a snack for the cray.
I have a tetra and 3 barbs with my cray, and they do fine, because they are faster than Charley.
What kind of Freshwater invertebrate would live peacfully with a Betta then?
:Bettas are very territorial and fearless, but he wouldn't stand a chance once locked in your crayfishes' claws. I hope you don't risk it, dear. Put the mudbug in a separate tank and provide crannies and crevices for him to retreat to. Larger tetras and minnows could be good companions for him, as could be a nice sized apple snail. However, it you want a companion for your betta, those little glass shrimp are harmless and amusing, and have most of the 'crust' attributes. (I've had them in my betta tanks for years and have found them to be a great little scavenger. Caution: they can 'flick' themselves out of the tank if startled! Good Luck!