As I think you already know, To increase the audio buffer size beyond what a particular audio driver allows, you need a different driver.
The closest thing I know to a 'universal' ASIO driver is Asio4all - you could try getting in touch with its author and asking if they could add another setting or two! As illustrated in piiperi's answer to your other question, most people's experience is that increasing buffer size leads to diminishing returns in terms of CPU overhead, hence why settings over 1024 or 2048 samples are not something most people feel they need.
In that question you stated that you aren't recording, just playing back. In that case you might be able to use another driver, such as the Windows WaveOut API. When I'm using this (as a programmer, through NAudio) I seem to be able to request very long buffers - e.g. over a million samples. Even the default buffer length (in my setup) seems to be 6615 samples. So it might be worth seeing what other driver types your audio software supports.
Another way to attack the problem would be to investigate why you are seeing a lot of performance difference between higher buffer settings, when it seems like that's not typical experience. It may be that the manufacturers of your DAW or soundcard could help here.
At some point, I wonder if bigger buffers would start making things worse, not better; you might end up in a situation where the whole address range for the buffer doesn't fit in the CPU cache and you start getting cache misses, which will distract the CPU more from its job of filling the buffer. This is just a 'musing', not something I've observed.