Whether you want the different recoveries to keep the constant hazard rate (equivalently probability of default, probability of survival) or constant mark to market (upfront) depends on why you're calculating this.
A vanilla CDS is usually quoted in the market as an annual spread (fraction of the notional), for example, 200 bps means that 2% of the notional paid every year. However the CDS is actualy traded with a standard "running spread", usually 100 bps, and an upfront fee. (Some distressed names are already 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 for the given running spread. 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 tread high-yield names often keep track of their own physical recovery assumptions $R_P$ different from the standard $R$ used for quotes, perhaps even having a term structure. This alternative assumption does not affect their market to market / upfront, though. They might do some calculations while keeping constant their mtm, such as derive hazard rates $h_P$ (having term structure) implied by their physical recovery assumptions $R_P$.
On very rare occasions the market participants agree to change the quoted recovery assumption for some name, e.g. from 40% to 25%. In this case they will aim to minimize the mtm impact from this change, while allowing the risk neutral hazard rate to change significantly. I don't see any use in looking at different recovery assumptions while trting to keep the risk neutral hazard rate constant.
Also, 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 could 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.
However I'm not sure if it makes sense to combine risk-neutral (used for quotes) and physical recovery assumptions like that. I'd be more comfortable if you had your own physical recovery assumption $R_P$, and kept constant the hazard rate $h_P$ implied by $R_P$, and then backed out spreads using alternative recovery assumption $R_P$
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.