How can I test the rotor winding of a single phase brushless alternator on a generator?
Update: Assuming the windings fuse and diodes are OK, it appears the rotor needs a current kick start to magnetize the coupling and thus self-excitation of this series wound brushless AC generator with DC rectifier option. There does not appear to be any regulation as this design excites the rotor with more load to maintain the output voltage,a sort of unity gain negative feedback magnetic amplifier.
It’s like a catch-22, if after not being used for a long time there is no Remanence left in the rotor to increase coupling to the stator, it can’t seem to start the current loop to generate voltage, as there are no permanent magnets. (Not familiar with series wound Motor generator series)
Normal alternators regulate voltage by linear current regulated grounding the excitation single winding.
Yours seems to be different with 2 excitation wires so DC current polarity matters. ( and the rotor diodes take care of that)  Voltage will be proportional to RPM and excitation current which requires more excitation with a load is handled by precise design of each coil and gap.
You can compare “load regulated Voltage with a fixed minimum RPM and fixed rated max excitation current. The ratio of load regulation in % will determine your alternator minimum impedance into a suitable battery (both stator and rotor impedances are coupled by the excitation so in series with Diode resistance x2)
If it is indeed brushless, then there must be 2 Hall sensors that go to 0V when flux is detected and pullup high otherwise. then you must externally commutator the stator windings in the correct phase using the Hall sense feedback.
Your “No Power” state may be wrong excitation current for a minimum RPM or a fault somewhere.  No need to look at rotor unless no fault elsewhere.  (See manual Note added below)
 this video makes it look so simple (: 
New info from your link:

Which not parts have you tested on this diagram? Fuse? Diodes?, Disconnected Cap? Which wire colours match? Mismatch?
Annotate above diagram , SVP.
Note it says:

If the generator does not self-excite, it should be excited by applying an alternating voltage of around 50÷230V to the capacitor heads for an instant..

The Cap is chosen to shift the line frequency toward 90 deg to make two phases from one.
Since the rotor has clamp diodes it will rectify the induced current to polarize the rotor excitation current waves.