A lot beginner barrel racers come into the game thinking you have to have specific stuff for barrel racing..and this is true, to a point.. ANY bit is a "Barrel racing" bit..
The tom thumb is a bit that can get very mixed reactions from a horse. Usually not so great reactions, especially in the wrong hands or heavy hands. I don't like them and I don't use them.... A good step to take from a tom thumb is the Jr Cowhorse like many posters have already said. I personally love that bit..
I understand why you said you don't want to put a chain in your horse's mouth..but what a lot of people don't realize is that there are VERY different types of chain mouth pieces..Some can be extremely harsh, like the bicycle chain mouth piece...Or be VERY light, like a light chain mouth on a hackamore combination bit..That said.. Just because it's a "chain" doesn't exactly mean it's harsh..You just have to know your bits and what they do.
Nikki rides in a snaffle mouth piece combination bit.
Hickory rides in a Jr Cowhorse, twisted mouth piece..
And Lark was ridden in a bicycle chain mouth piece before I got her and I refuse to put THAT type of chain back in her mouth because she is very sensitive about her mouth now.. So she rides in a LIGHT chain piece hackamore combination bit..
This is one of the harsh chain bits..