Here a start for you to reasearch with basic info. My concern is mixing shrimp and a betta's as I think will eat the shrimp is they are too small. I never housed bettas so I don't know about there temperment, but I know fish think shrimp are a taste snack. SO go with the larger shrimps would be safer.
Red Cherry Shrimp - Neocaridina Denticulata Sinensis - Size: Up to 3/4" pH:6.3 - 8.2, Temp: 70-82, Origin: Taiwan
Amano Shrimp - Caridina Japonica - Size: up to 1.25", pH: 6-8, GH: 8, Temp: 7-28 C, Origin: Japan
Malaysian Rainbow Shrimp - Caridina sp. Rainbow - Size: up to 1" pH: 6.5+7.5, Temp: 20 to 25C Origin: India
Brazilian Yellow-Belly Grass Shrimp - Palaemon Pantanal - Size: up to 1.5" pH: 6.4 - 7.8, Temp: 48 to 89F can handle more extreme cooler temps as long as they are gradual, Origin: Brazil
Rudolph Red-Nosed Shrimp - Caridina gracilirostris - Size: up to 3/4" pH: versatile Temp: versatile Origin: India
Bumble Bee Shrimp - Caridina trifasciata - Size: 1/4 up to 3/4" pH: versatile Temp: versatile Origin: Japan
White Spotted Pearl Shrimp - Macrobrachium mirabile - Size: 1/2 to 2" pH: versatile Temp: versatile Origin: India
Bettas differ in character. There are some that even leave Red Cherry shrimp alone, and then there are those for which they're only a snack. Plus, you never know when the first type becomes aware that shrimp are a nice snack...
My Betta (imbellis, not splendens, they stay smaller) eat the occasional cherry shrimp if it fits into their mouths. And if it doesn't fit it is made to fit - bitten apart . However, as Cherries reproduce in high numbers and find many hiding places in that tank their number keeps increasing despite the occasional snack.
To be on the safe side pick some larger shrimp, like Amanos - if your tank is large enough.
I would not recommend e.g. Atyopsis moluccensis, as they need a strong current in their tank, which the betta dislikes.
what i would say is go to your LFS see what shrimp they have. If they have ghost shrimp (usually really cheap) buy two or three see how the betta deals with them, If they are ok they go ahead and look for a prettier shrimp (cherries are my personal favorite).
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