Others have correctly pointed out that the "resolution of identity" method by itself is just used to help obtain approximate energies within a particular basis set. Indeed, the abstract of this paper discusses the RI-MP2 approach, and says that the RI approximation resulted in approximate MP2 energies that were within 60 milli-hartree of the true MP2 energies (over a set of more than 350 molecules). More details can also be found here.
What you're describing in your question, about stepping up by "one zeta" in convergence to the complete basis set limit, is something slightly related. For example, the abstract of this paper says that 4Z quality energies can be obtained by using a 2Z basis set when using "the R12 technology". The first paragraph of the introduction section also says that you can get 5Z quality with a 3Z basis set. The R12 technology requires evaluation of 4-electron integrals, which would ordinarily be very difficult, but are made simpler with the resolution of the identity approach (see the introduction section of this paper for example). This attempt at obtaining energies that are roughly equal in quality to the energies that you would ordinarily get with a higher-Z basis set, is also exemplified by methods such as RI-MP2-F12, which you can see is implemented as described here.
I'll point out though, that a lot of people say things like "explicitly correlated methods like R12 and F12 gain you two zetas" without understanding that this "two zeta" rule is more of a "rule of thumb" than a rigorous theorem. Observations in some datasets suggest that you can get 4Z quality energies with a 2Z basis set, or 5Z quality energies with a 3Z basis set (as described above), but this does not mean that you'll always observe a 2Z jump in quality; and the word "quality" is also open to interpretation. Some people might observe a 1Z jump (as you described in your question), and some people might observe less than a 1Z jump (i.e. barely any jump at all), and some people observe something closer to a 3Z jump in some cases. This "2Z jump" has been observed mainly for obtaining 4Z quality with 2Z basis sets and 5Z quality with 3Z basis sets, but not as much (or at all) for obtaining 7Z quality with 5Z basis sets (for example), largely because (for example) the auxiliary basis sets needed were not available to do such 5Z calculations. People have said that the higher you go up in Z, the more benefit you'll get from these explicitly correlated methods (like R12/F12), but what they don't usually seem to realize is that when you're doing calculations with a high Z (for example 5Z or 6Z), the reason is to get high-accuracy results, and the approximations associated with using these R12/F12 methods can outweigh the benefits that they provide in terms of approaching the CBS limit (and the lack of RI auxiliary basis sets for higher Z values can also be an obstacle). R12/F12 methods often in fact overshoot the CBS limit, meaning that they often give energies that are lower than the variational principle would allow. This shouldn't be a problem for you in this particular case though, since you said that you're using double-hybrids (which means that your error in the electron correlation treatment might be bigger than your basis set error anyway), but it's something to be mindful of when people tell you that you can get an 2Z jump for an arbitrary calculation by using an R12/F12 method.
I'll finally point out that since your question points out that Gaussian, Q-Chem and NBO didn't work out for you, it may be worthwhile for you to try mrcc which is available for free and very good for double-hybrid calculations, and calculations on huge systems (see here for example).