Stuff is coming together. I can switch the receiver to ublox binary mode, configure the PPS output and start a self-survey. PPS signal is nice and steep, somewhere in the region of <4ns rise time. Here ends what my scope can resolve. I can also parse a couple of common messages, like, clock, nav, tp, svin, status, etc. I think that's enough for now.
With the TP message I should be able to perform sawtooth correction, but then of course I can forget about the 128PPS averaging. I'll probably try both and post some data.
PS: not convinced that 128PPS is a good idea, I took a look with the scope and I see very frequent 20ns jumps from pulse to pulse.