How do I manage the power draw on these custom light pieces?

Total load depends entirely on how many of those LED modules you have and what kind of power draw they have individually, but for a display, you want to avoid putting all of them on a single run of 240V lighting. If you hook them up in series with the traditional string, you'll draw maximum current from the source and the integrated circuit in the singing bulbs is likely not designed to handle that continuous draw, which can cause the internal components to fail or trip a breaker. The safe method is to treat each singing bulb like a separate power consumer and run them off a dedicated power strip or a separate branch circuit if your house panel allows it. This prevents the load from overwhelming the existing Christmas lights. Since they are LED components, thier heat generation is minimal compared to old incandescent bulbs, so while PETG can degrade if overheated, the primary risk here is electrical overload, not melting plastic. You need to calculate the Amp draw for one singing module and then multiply that by the total number of units you are running. If the individual bulbs pull a low amperage, say 0.5 amps each, and you are using 20 bulbs, your total draw is 10 amps, which is easily handled by standard extension cords. If the individual modules draw more than a few amps, you definately need more than one power strip to stay safe from a circuit overload. Use a heavy duty extension cord rated for the total required current, and plug that into a breaker that can handle the collective wattage.

You got these 3D printed things, but don't forget about the run of cable itself. If those bulbs are scattered all over your yard, the distance matters just as much as the draw. Even if the amps are fine, you'll start seeing a voltage drop if the gauge on the wires is too thin over a long stretch. It is frustrating when new stuff causes issues that old guys solved with a thicker gauge and a good splice.

Make sure whatever connection you use to tap into that main string is rated for the distance. Using a cheap splices will just cause headaches down the line. Keep it simple and use quality hardware.

I always worry more about the physical connection grounding than the total amperage itself. Even if the power draw is low, a bad ground can make those sensitive integrated circuits fail prematurely. I once spent a whole weekend tracking down a fault caused by a damp connection that slipped past the main disconnect. For any DIY setup, I require a dedicated GFCI protection right at the outlet source. On my biggest runs, I always set the breaker to 15 amps minimum to ensure stability, even if the bulbs technically only use half that.

I think the physical connection points are just as critical as the total amps. When you have small LED modules spread out, you end up with a lot of connection points that need to handle the current safely. I ran into this same issue when I was wiring a whole string of custom displays, so I started using screw terminal blocks instead of just using butt splices. It realy made a difference in preventing any little hotspots at the connection. Makes the entire setup look way cleaner, too, so you don're not worried about some dodgy connection failing down the road.

I’ve been dealing with vibration fatigue in my own custom outdoor displays, and I find that the movement of those small rotating components you mentioned, specifically the eye mechanism, could introduce mechanical stress that far exceeds the electrical considerations. Even assuming the power consumption is meticulously calculated, the constant physical oscillation inherent in integrating these animated faces into a larger, likely vibrating setup, presents a risk to the structural integrity of the PETG housing over time. Have you considered introducing some form of vibration dampening material inside the bulb, or perhaps mounting the entire unit on a slightly resilient substrate, to prevent undue wear on the gears and connection points?