A vanilla CDS is usually quoted in the market as an annual spread (fracion of the notional), for example, 200 bps means that 2% of the notional paid every year. But when the CDS is actualy traded with a standard "running spread", usually 100 bps, and an upfront fee. (Some distressed names are quoted on upfront.) In order to convert the quoted spread to upfront (which is your mark to market), you need to agree on the recovery assumption with the counterparty. Usually everyone uses standard recovery assumptions - 40% for corporates, 25% for emerging markets, etc. The quoted $S$ and $R$ together convey the information about how much upfront fee you'd pay / receive. The ISDA standard CDS model also uses an IR curve, but the it changes the numbers very little.
But no one knows what the recovery will be if a credit event actually happens. People who reade high-yield names often keep track of their own recovery assumptions $R_A$ different from the standard $R$ used for quotes. This alternative assumption does not affect their market to market / upfront, though. They might do some calculations while keeping constant their mtm.
In the past, some people used to trade a "fixed recovery" variant of CDS. They have been pretty much dead after 2008.
So, let us consider your example: you observe in the market a vanilla CDS quoted as $S$ bps, assuming standard recovery $R$, and you want to derive a quote for a fixed-recovery CDS, contractually specifying fixed $R_A$ recovery. The most common $R_A=0\%$ means that after the credit event, the protection buyer gets the full notional and does not need to deliver the defaulted bond (or its cash value) to the protection seller. But I've seen fixed non-zero recovery contracts in the past.
Only as a starting point, you could say that the spread is very approximately $≈S\times(1-R_A)/(1-R)$. Note that in this example you're using both the contractually specified physical recovery $R_A$ and the risk-neutral recovery assumption $R$. Note also that if $R>R_A$ then this spread is $>S$ because this contract is more valuable to the protection buyer than the vanilla contract, and vice versa.
For calculating mid in order to mark to market, you'd calculate the risk neutral hazard rates (or neutral default probabilities; but not upfront) from $S$ and $R$, and use the risk neutral hazard rates to back out the $R_A$ spread (or $R_A$ upfront) using $R_A$ recovery assumption. The ratio you cited is a rough approximation of the spread obtained by keeping the hazard rate constant, rather than upfront constant. Note that the upfront and mtm of a fixed-recovery contract will have much greater sensitivity to $R$ than that of the vanilla contract.
But the bid-ask spread on such an exotic illiquid product would be so much wider than the vanilla CDS that none of these details might practically matter much.