Going into Quant Finance without grad school?

Your background looks good enough, but due to the current crisis, lots of PhDs hang in the street, it is not easy for a even master student to get a job as a Quant, to be honest. Therefore my suggestion is to try looking for a quant trader instead, which normally requres a bachelor or above, good luck. Take a look at the quant job board and see if there is a job what you want.