Electrical – How to measure very low frequency noise

Noise is difficult to measure, and the amplitude you see on your your scope is only a first indication of the level.
Do you want to measure absolute noise levels, or just comparative? In the latter case the scope could be a good instrument, but at the given levels the average $500 scope will have so much noise itself that any measurement becomes in fact meaningless. You need a high quality scope + ditto probes to do this.

The difficulty with measuring noise is that it has a wide bandwidth continuous energy spectrum (the continuous spectrum makes it hard to separate noise from signal, notch filters may work). Ideally you measure the noise energy through RMS-to-DC conversion. This is not for the faint-of-heart, as your RMS-to-DC converter has to be very sensitive due to the low levels, and wideband. And of course be low-noise itself! Liquid nitrogen helps :-).
In any case, absolute signal-to-noise ratio is not as easy as reading amplitudes.

I've tried a similar setup and I've realized that you're using a very small sweep time (500 us). Your transient, that seems so wide in the picture, last only about 30 us. When using a quasi-peak detector it will be certainly removed.

I can show you my pictures. In my setup I've used a 10 dBm signal, 868 MHz central frequency, and an offset of 200 kHz when looking for the transient. In all the figures RBW is 120 kHz and VBW is 100 kHz.

Here it is the zero-span picture, no PA ramp is used:

My sweep time is 5 s. The pulse is then about 0.8 seconds wide, and the transient is less than a few ms (didn't measure it accurately).

When using a Quasi-Peak detector I get this:

If I repeat both measurement after activating PA ramp in the transmitter, the transient peak disappears:

And using a Quasi-Peak detector:

Conclusions:

Your phase noise is in fact fine, and so are the transients. The fact that the transient doesn't change when enabling PA ramp seems to indicate that there is some kind of implementation error with this feature. Anyway your transient will strongly smooth out under a Quasi-Peak detection.