Good luck with your search.
I have heard the Koetsu RSP and the Lyra Kleos with observations similar to yours. You might consider the EMT JSD S75 (75th anniversary edition), which is a cartridge I purchased recently. Great detail without thin bright sound, natural voices, and bass with drive. In your stated price range new. Tone Imports is the US importer. The link below has some information on EMT.
Good luck with your search.
I have a Kleos on a JMW-9 signature tone arm and an Audio Technical AT150ANV on another JMW-9 signature tone arm and the AT is much more liquid than the Kleos and it tracks much better as well, especially the inner groove.
The AT is one of the best cartridges I've had the pleasure to use and dispels the notion that MC cartridges are superior. I would put it on my top 3 picks.
Benz Micro LP-S mr. Ive owned both the Kleos and RSP. It has all the weight and texture of the RSP, and close to the kleos detail without ever sounding tipped towards the top end. Check reviews online. Its a powerful, detailed, sound, balanced from top to bottom. Mine went around 4 years and thousands of hours before retip.
FYI on the Benz Micro LPS. My understanding is all new production has the micro-ridge stylus. You could confirm with Musical Surroundings, the distributor.
“First, yes, there has been a changeover with the LP S. Benz has gone to a micro-ridge stylus (thus the “MR”). Second, so far I think the Benz LP S-MR is one of the two or three best cartridges I’ve had in my system. It is outstanding.” – Jonathan Valin, The Absolute Sound
The Benz Micro LPS mr is still in production
"Benz Micro cartridges have been in production since the 1980s, earning recognition throughout the world as one the top brands in the field. Ernst Benz, founder and designer of the original Benz-Micro cartridges, retired in 1994 and sold the cartridge production to Albert, his close associate and designer of the Benz-Micro PP-1 phono stage. Albert relocated and modernized the factory, employed a highly skilled workforce, introduced many new models, and expanded his OEM cartridge manufacturing.
In the past year, Albert has stream-lined his factory, focusing on the most popular high-end Benz models: the LP-S MR, Ruby Z, Gullwing SLR. "
Have you tried tuning your step-up transformer? I use a bank of nude Vishay loading resistors (on the input), and switch in the loading that sounds best for the given record. Then I use an air gap capacitor to tune the output from the SUT.
My RSP sounds plenty lively in my system, but certainly sounds dull if loaded too hard.
I use Sowter transformers as per their recommended schematic, with lower values of both resistance and capacitance in series across the output. Parallel to that is a range of possible load resistances from far above to far below what is recommended. These are inserted across the outputs of the transformer. Trial and error showed the way here - obviously, highly system dependent.
My phono stage (home-brew) has airgap variable capacitors (100 to 400 pF, but will be changing to 0 to 1000 on the next iteration) across the inputs, i.e. the outputs of the SUT’s. These serve to fine-tune the sound, in a slightly different sense than the loading resistors - it’s hard to describe, but the airgaps seem to me to provide a useful sonic complement to the loading resistors. YMMD, but it may not be appropriate even to attempt it, in your system, as less capacitance is brighter. Come to think of it, have you tried ultra-low capacitance cables?
Have you tried pushing the load resistance up to 1K, or even higher? Say 10K? Loading with a transformer is a black art, as you know, and is, dare I say it, highly system dependent?
I found my KRSP to be highly sensitive to VTA and azimuth. But, of course, you will already have explored all that.
I spent so much time on my KRSP because I had to make it work - it is a considerable luxury for me.