I commend you on your desire NOT to hack the original wiring. IMO "interfacing" is better - ie, using mating connectors as you have done and adding whatever wiring etc is needed.
Apart from not hacking the OEM setup, it makes for easy removal should that ever be desired.
[ I drive 1960's vehicles. Though some of the original
interfacing was tedious, that has been paid back multifold with subsequent "plug & play" transfers! And other collectors are happy to get my vehicles in OEM condition - not that I've ever sold any. ]
In fact IMO any
conversion of headlights to relays should be done by controlling the relays from the OEM headlight connector. However converting to a dedicated low beam has an additional hi-beam flasher consideration, but I'll cover that below.
Now, what you want sounds like the mod I did to my 4 headlight vehicles. They were originally the classic Japanese hi/low outers and high inners.
I decided I wanted dedicated low beam outers (whether they were hi/low shells or dedicated single filament low beam shells).
The solution is simple.
Instead of powering the low beams thru the dip switch, move the low beam signal or power to the beam power - ie, the beam switch position that sends power to the hi/low dip switch.
(Do you see the "logic"?)
The hi/low switch essentially becomes an on/off switch for the high beams. And the lows are always on if the "beam" switch is on.
When I say "power" above, I mean whatever signal is used. I have always fitted relays for my headlights, hence removing high currents from the switches (and having brighter headlights - being of the non-HID variety).
But even without relays, the
wiring swap should not be an issue since the beam switch has to carry whatever current is required for the beams, or their relays if fitted as OEM.
And FYI - any beam switch signalling polarity difference can be handled by the relays - eg, maybe ground switching switches +12V to the beams, and sometimes (though AFAIK not common), +12V switches might turn on relays that connect GND to the filaments.
Now for the catch...
The high-beam
flasher (or pass/overtaking and police warning) will only flash the high beams. That may be undesirable or even illegal.
That too is an easy fix. Run a diode from the high beam signal or power to the low beam (line end towards the low).
Assuming relays and +12V switching, run the diode from the +12V
signal input of the high beam relay (86) to the input (86) of the low beam relay.
I suggest also fitting a diode in the low beam signal to the low beam relay to ensure that hibeam +12V does not backfeed into the "beam" circuitry. Usually that isn't necessary, but just in case.
The diodes to the relay coil(s) can be any of the 1N400x series (I suggest 1N4004 else 1N4007 if you get some as they can be used for spike-quenching of relay coils plus a host of other automotive wiring applications. Hence get a pack of 10 etc.)
The 1N400x handles 1A of current which is sufficient for typical automotive relays. (They are usually 250mA at most, so a 1N4004 etc can handle up to 4 relays.)
Alternatively the hibeam +12V can come from the hibeam relay's +12V output if that's easier. (One could get into a failure-mode discussion regarding this... 86 to 86 is better if the hibeam relay fails, but relays are so reliable... Since I run a relay for each filament, I have the 2 sets of diodes - one per side - hence if one diode fails, I still have the other diode to provide the flash signal.)
Just remember - the lines on the diode (for both the hibeam flash signal and the feedback blocking/isolation diode) BOTH point towards the +ve of the low beam relay coil (86).
This reply got lengthier than I anticipated, but it should be a fairly complete guide.
Alas if you have beam sensing stuff (bulb failure warnings), then that's another issue, though that is an independent fix to this installation.
But I hope I haven't confaddled you with all my guff. It is easy to
picture with a simple diagram.
If anything is unclear or complex, please - just ask.