So I had a bit of a play. I've generated my own "super" saw and "natural" saw as well as recorded two of my synth's saws for us to look at.
I think these two types are the most important because they are different enough from the regular saw and would save voices trying to produce them from the regular SID.
I generated the "super" saw by combining a base saw wave with one at -5 semitones (and -3db) and another at +7 semitones (and -3db). I may have reversed the result to make it more similar to what I was getting from my synth. I forget. The sound is much the same as I tried both. It would be easy to generate from what the SID can already do.
I generated the "natural" saw by combining a saw and a sine. Note: Just looking at it again after I've already done this, its better to combine the sine at -3db as you get a closer result to the "Digital HD" saw (a "natural" type saw) from my synth.
I can't generate a triangle waveform with Audacity so I couldn't test to see what the "natural" result might be if we stuck to what the SID can already do.
I think these two saws plus a better white noise would be fantastic and would bring the SID up-to-date. It would be the equivalent of having more voices. Having the pulse and saw combination you suggested and the ability to couple the filters would make this SID version quite something.
My Natural: https://1drv.ms/u/s!AgcNCnteKUeegRF4zGYj2vuOr8vW
Synth's Super: https://1drv.ms/u/s!AgcNCnteKUeegQ9Z5i1NfXC8n12L
Synth's Digital HD: https://1drv.ms/u/s!AgcNCnteKUeegRBVFcxGqbv51Zan