KUALA LUMPUR, Malaysia—After developing a personal liking for nonalcoholic beer, businessman Ruzi Shuib took on a challenging task—persuading fellow Muslims here to try it.

It seemed a natural selling-point: Muslims make up 60% of Malaysia’s population of around 30 million, and are prohibited by Islamic dietary laws from drinking alcohol.

However, the market for nonalcoholic beer has been slow to catch on in Malaysia. Many…