@Anakchan:
an LC-FIlter is the combination of an inductance and a capacitor, the (C)LC-filter stages can be seen on the internal power supply board (this two electrolyte capacitors with added small capacitors together with the inductance inbetween (this black .104 and .221. parts). they work as low pass filters to kill hum/noise and remove the switching frequencies of the dc/dc converters from the power lines.
i think the easiest solution for you could be to just add a small resistor consuming 25 mAmpere directly connected to the filament / heater pins (pin1 and pin7) of your SF tubes, so when inserting the SF tubes the internal power supply will "see" normal 50mA tubes and has to deliver 50mA per tube so may be more stable then:
i have nice and small 7pin tube adaptors that also could be used to connect a resistors and a tube socket to so only two adaptors with resistor and socket instead of soldering something to the tube pins (just 75 european cent per piece + resistor + socket):
resistor should be 1.4 volt / 0.025 Ampere = 56 Ohm.
this should work if the unstability of the DD power supply is related to the too low current (in fact the filament power is provided by an integrated circuit DC/DC Step Down Converter).
if the unstability is related to the smaller inductance of the smaller SF filament this may not work.
for my modded amp i in fact have added additional LC and RC filter stages on two boards to be connected between internal power supply and tube amp internal power connector, should be very easy to make a small additional board with an additional LC filter stage for the tu-06 as there is enough space, so connecting the five pin out of the internal power supply board to such a board and then providing the same connector to connect the added filter board to the tu-06 amp (as i have done for my modded tu-05).
maybe also shikada san can provide you with such a small filter PCB board just adding the needed LC filters so you could just connect the board between internal power supply board and amp, so your Option 2 can be prepared by shikada san without sending the amp to him!
okay, i will modify two df96 tubes with a resistor to test and post a picture here for you to see how this could be done.
you could ask shikada san to test if a 56 ohm resistor in parallel to the filament pins of the tube(s) is enough to solve the problem.
this modification can be done by directly soldering the resistor to the tube pins.
i will check the JRC2392 step down converter data sheet if there is any recommendation for LC filter for low current use.
btw, just for clarification:
Tube: DF91, 1T4, 1U4, DF904 - 50mA filament current (no problem)
Tube: DF96, 1T4T, 1T4SF (DF97, pin 4 and 5 connected) - 25mA filament current (problem)