Alright, I think that's what I'll go with.

It looks like LM8560 style output, so with the output generating the 900Hz signal, gated by 2Hz to generate the beeping...
In the radio control path there will be some kind of filter, turning the 900Hz into a solid steady voltage. You may need something similar for the relay. The time constant depends whether you want the light solid ON (then it needs to cover more than 0.5s) or flashing (cover just the 2ms).
Example is drawn in the page 4 of the LM8560 datasheet (assume it is its clone or something with similar output).

Off topic for a sec: I'm into sportcars, muscle cars, various trucks and motorcycles. I do own a custom Hyundai Genesis Coupe that I've owned since 2021.
Not a daily car by any means, but certainly fast and fun, when warranted. (Racing, shifting through the gears, insane turbo sounds on the highway... etc.)
What BMW do you have, @SussexEuroSOX?
Back on topic: I did not get the Sweet Potato Pie, but I will likely be on the journey for one, tomorrow morning, taking the Genesis. I've never tried
the Sweet Potato Pie from Meijers, so I'm interested in going there tomorrow to pick one up and try it out, reporting my results here.
Pumpkin pie also sounds delicious right now as well, also with whipped cream.

The Safelines are much harder to find than the Clearbanders. I’m sure if the Clearbander you want sells, another may come up again eventually. For the $6.30 difference, I think the Safeline listing is a better value. The fully coated Sylvanias really aren’t bad, and the T lamp is like a bit of gravy.

In fact, S102/202 optos I mentioned have forward drop of less than 1.5V @2A, so still may be OK even for flashing 12V halogen directly, especially accounting for somewhat constant-current behavior of small power halogen transformers.

To invent an optimized circuit, some hint of actual clock schematics is needed, @lightsofpahrump was asked for, but he prefers to keep silence. Ok, his choice.

The Alarm Output pin is going through the MODE selector to either the buzzer or the transistor that switches on power to the radio IC depending on position and there is a turning-off-the alarm link in the switch I explained in my last post that shuts down the alarm mode in the chip if the mode switch is not set to an alarm position. I want to tap the output of the chip directly, triggering the relay through a MOSFET to avoid overloading the IC.

In these things (judging what I have at home) is just one single transformer with one secondary feeding the clock/radio part and then the 12V high current one feeding the halogen. All the switching is then on the low voltage side. So triac would have way too high voltage drop when starting from 12V.

I think the only way to proceed is to trace the buzzer and radio power control circuit and start from there.
To control the radio, there already is some kind of detector/rectifier, as the radio does need a permanent power. And the clock IC very likely provides the same signals all the time (assume one of the clock ICs, not the microcontroller), they are only afterwards blocked by the mode switches.
So depend on how that is arranged, you may either tap on the existing control (if the radio/buzzer switch is after that) or replicate the radio switch circuit to control the relay.

The LM8560 generates only "buzzer" alarm signal by a single PMOS open drain output (it uses PMOS logic technology, so the ground of the clock circuit is positive), so there is a kind of filter circuit which turns that into continuously (when the buzzer is buzzing) available power for the radio.
The is a second "sleep" output, but that works only for the "sleep" function (press the button, it runs the radio for set amount of time and tyen shuts it down), these are combined in the radio power control circuit, but you won't need that (unless you want tye "sleep" function for the lamp asc well). This "sleep" usually goes around the alarm mode switch directly to the radio.

You are spot on about the transformer!
I want to turn it on with a relay driven by a MOSFET and I found the alarm trigger pin on the schematic. I could tap that, because judging by the schematic, the alarm is disabled by a link in the alarm/radio mode switch sending a signal to the chip unless the switch is in AUTO(radio alarm) or ALARM(buzzer) position, not in ON(radio on independently of alarm) or OFF(alarm disabled and radio off).

Well since we are bringing this back I might as well chime in. Autistic among other things.

More about what I got here:
https://www.lighting-gallery.net/index.php?topic=18186.0

More about how I feel about it here:
https://www.reddit.com/r/offmychest/comments/1q64s2s/i_feel_like_i_am_better_than_everyone_but_i_know/

Of course all the eye contact and social situation shenanigans common to many people on the spectrum. But more interestingly I now know that autistic individuals are apparently a lot more likely to have a bunch of other neurodiversities like I got, among sexuality differences as well, which in particular I find very fascinating. Could be another minor hyperfixation, idk. I sort of want to research more into the biological aspects of autism, but lighting is more interesting to me right now lol.

As to the rapid change of interests/hyperfixations, I can totally relate to that. It can be fun, but more often than not it is just annoying. I have an electronics project that has been going on for over a year now, and it is taking so long because a lot of the time I am just not interested in it. Totally unpredictable.

Twenty one pilots! They are my favourite band! Their best song is (in my opinion) Centre Mass

Yup, I’m autistic. OCD, ultra sensitive hearing, Anxiety, depression and occasional panic attacks. It’s not all bad though: as soon as I here a song with drums, I can almost immediately copy it on to my drum kit, I’m always quite resilient (apparently), insane attention to detail and I have very powerful visual thinking! One good thing I don’t have is being very smart. I’m as dumb as a rock! But, I remain quite quiet about it, for I do not like talking about it in casual conversation.

I need to pick up shopping from Carrefour tomorrow morning, I’ma take my BMW Dominoes Is actually delicious though! I love their Margaritas! I am a very picky eater, I only really like Margarita pizzas. A go to place in Europe is definitely Belgium (and I’m not just saying that cause I lived there for 5 years and am currently there with my family, again) if you like food, Belgium has you covered! If you like cars, Yep, Belgium. If you like street lights, Belgium again! Anyway I just had a sleepy bed time snack, A weird small Nutella covered Waffle thing! I also found some Marshmallows!