Part of the point of using real religions in a RPG is just like any other cultural product - you don't have to provide an encyclopaedic explanation of what you're doing, precisely because you are locating your work within a larger tradition that audiences/participants are already familiar with and themselves located within.
Well, that's sort of my problem here. Because as soon as you start looking at
other things, you get into some much more hairy conundrums. For example, consider works like
Neon Genesis Evangelion or even
The Chronicles of Narnia. The former is...shall we say,
extremely loosely inspired by real-world Gnosticism, and has Christian and Jewish symbols and mysticism sort of thrown around willy-nilly. It's also very specifically created in a culture for which Christianity in general (and Gnosticism especially) is foreign and unknown. The latter is
often upbraided for its depiction of a really obviously Arabic-like culture that worships the
explicit equivalent of Satan, and despite my love of the series I can appreciate this criticism.
Even the OP has some issues in that direction, like referring to Shiva as a Buddhist deity--rather clearly an out-of-context thing. And that's ultimately the problem with a published game that includes any deeply-impactful elements of a religious tradition: it's gonna go far beyond the "larger tradition that audiences/participants are...themselves located within." Not, I should note, that any given element
can't be deeply impactful, but as Asisreo noted, retellings and pastiches of things like David and Goliath, the story of Momotaro or The Bamboo Cutter, heck even some of the Thousand and One Nights, are things you see these cultures doing, whereas retellings and pastiches of--say--Moses receiving the Ten Commandments or Muhammad transcribing the Quran are
substantially thinner on the ground. The Narnia retelling of Christ's sacrifice is probably the only example I can think of that does such a thing, but that was done by one of the most prominent lay theologians of the 20th century. Not exactly comparable to a (frex) Christian or atheist/agnostic caucasian American writing an adventure module that features totally-not-Amaterasu needing to be drawn out of a cave, or having that's-so-not-Raven tricking definitely-not-Bear, or whatever.
Which is to say: Though I'm not keen on cries of "appropriation," there is an EXTREME risk of being cavalier, even by totally innocent accident, with really really important values or stories for other cultures, often ones that are poorly-represented and whose actual members rarely get a voice in how they're portrayed. That sort of thing sucks, and it's bad to contribute to it. Not saying it would be
impossible--as I said before, Islam is a major influence on the fictional monotheistic religion of my home game--but that you have to be
really really cautious because of how likely it is that your words
aren't going to be seen and understood in a context shared by both you and your audience.