...it seems you’ve already answered your own question.... and the switch is at fault.
The fact Hi Beam (ie the Blue wire) illuminates the dipped filament tells us that the switch unit is receiving quality input power, ie the fault therefore lies at the Lo Beam switch terminals or downstream of...
But then you go onto say that the Lo switch also cuts out under load... ie the white wire & connectors beyond here and the bulb holder are likely ok.
So, for my money, you have a faulty switch unit, specifically in the Lo beam switching and somewhere within the area between the two red circles in the attached....
Q). Where were you measuring the switch fault... at the end of the switch loom (ie in the white wire within the rubber boot under the tank) ... or somehow within the switch itself?
As a starter for 10, spray switch cleaning fluid (or WD40) into the switch unit and exercise the switch a dozen times... in the hope of ‘cleaning’ it.
Also thoroughly clean / check the white bullet connectors within the boot under the tank. (upper red circle on the attached diagram)
If unsuccessful, you might then be looking at removing it from the bars and dismantling it... to clean the connections and / or check for a fray / break in the white wire where it wraps around the steering head.
Good luck...
Simon
PS - if you do dismantle the switch take care and be methodical - they’re sods to get back together. Do your research... lots of good stuff out there... eg Martin_UK’s thread.
www.sohc.co.uk/index.php/topic,7533.msg51016.html#msg51016