I thought the comparator would be sufficient for the LDR setup
It is. Your comparator works just fine. This thread is discussing how to modulate (switch on and off) the power to conserve coin-cell life. Sorry for the tangent discussion, which I finish below.
What a log amp could do is make it easier to set a threshold level for switching between full daylight and full darkness, somewhere between dusk and dawn for your application over a wide range of light levels. Your current design works fine, but the voltage divider resistors that set the switching threshold were determined by trial-and-error. Not much illumination lee-way is provided with a linear potentiometer in series with the voltage divider, nor does that much accommodate variations among LDR sensitivities if you need to build more than one.
A log-amp would compress the huge range of LDR resistance variations that occur as a function of light intensity into a linear span from which you can easily pick a threshold level from full daylight to pitch black darkness. Of course the span isn't really linear, it's logarithmic, but the effect is to "linearize" the LDR response.
I think it might be worth your effort, from an educational point of view, not for your Halloween decoration, to at least breadboard a simple logarithmic amplifier circuit and "play" with it. You could start with circuits that use just one BJT and one op-amp to do this, but of course simple circuits are extremely sensitive to variations in temperature.
LDRs aren't the only devices that benefit from a log-amp. Have you ever been annoyed when the TV program switches to commercial and the sound level appears to double? A log-amp can help to fix that and restore "normal" sound levels. I am still working on a reliable method to automagically mute the sound when commercials appear, and there are probably others who claim to be able to do just that. My solution, so far, is to just watch programs on streaming media like Netfilx and Amazon Prime or watch Blu-Ray DVDs.
This is what makes it difficult for a hobbyist to use these devices as a simple switch, there is a lot more going on than a simple on/off action!
That is true, there is a lot going on "under the hood" but you don't need to know
everything yet to get moving down the road. The thing that distinguishes most hobbyists from "professionals" is we have the luxury of having time and materials to experiment and learn new things. In a commercial environment, time and materials cost money and have to result in profit. Hence there is a strong incentive to have the knowledge to "get it right" the first time when designing "new" circuits for commercial use. That's why engineers spend big bux for an education, and employers spend years exposing new graduates to the real world.
You, OTOH, have the "free" time to "play" with an assortment of inexpensive BJTs, choosing bias, choosing different voltages to drive the base and collector terminals, and choosing different resistors to limit the base and collector currents, while observing and measuring (and hopefully making journal entries of) the collector-emitter voltage (the thing you want to switch), aiming for no voltage drop when the switch is "on" and dropping all the voltage available from the supply across the collector and emitter terminals when the switch is "off". You may not reach those exact goals, but you will learn a lot about BJTs along the way.
Personally, at the low voltage you get from a single coin-cell, I would try to do this "switching" with a MOSFET. Unfortunately, if you just substitute a MOSFET for the BJT, you now have a source-follower and you have to make the gate voltage for turn-on larger than the output voltage by whatever the threshold voltage for MOSFET happens to be. And you need to find a MOSFET with not too much rd(on) resistance.
Maybe it is time to consider using a small electromagnetically operated reed relay.
You can bias a normally-open reed switch with a permanent magnet so it latches on, allowing you to remove the coil excitation. Later, to turn it off, reverse the coil current (or use another coaxial coil) with a pulse to release it from the latched state. The magnetically latching reed relay draws zero power except when changing states.
Hop