: Is it correct to write: "Your payment's being processed"? Is it correct to write: "Your payment's being processed" ?

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Yes. But you probably still shouldn't use it.
The apostrophe can (and in this case does) represent the contraction, but as motosubatsu mentions, it's ambiguous.
Also, from a practical point of view, the point of using a contraction is presumably to make it shorter and easier / quicker for people to read. But in this case, people will have a sort of muscle memory for the phrase 'Your payment is being processed', and if you change that it's probably it will actually take them longer to read, as they have to process it as new information, rather than stock information.
The risk of this delay, considering you've only gained the removal of a single space (because you've only removed one letter, and it's been replaced with a punctuation mark anyway), makes it seem like this is unlikely to be worthwhile.

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No.. while I see that it could be interpreted as a contraction of "your payment is being processed" apostrophe-s after a noun is indicating a possessive

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