Juan -- a couple of
thoughts that came to my mind based on your blow
-by
-blow report:

You might command a move off 435.000. As I recall, the
synthesizer reference is at 10.000 MHz. I could conceive an internal
spur generated by the reference oscillator.
The 6.8 kHz spur interval is certainly weird, but it sounds
like it might be due to the synthesizer chip's RC low-pass filter with a time
constant ~ 1/6.8 kHz = ~140µs. Is it possible that something inside the
loop is oscillating? Try a scope probe on the VCO's tuning voltage.
On topic #2, is it possible that you can see microphonic on
the tuning voltage?
I'm interested to see that you also find the IQ as a very
useful piece of test equipment. Here is another test you might want to
try: Remember that the IQ does down to (nearly) DC. You could try using
it to look at the spectrum of the synthesizer tuning voltage over the ~0-20 kHz range.
If the chip is locked, there should be very little noise; but with the
spectrum you are showing, I wouldn't be surprised to see noise that
mimics the 10.7 MHz power spectra you show 