So for anyone who may be curious, I managed to get the Child Trait part working as desired (Vamp Run I'm still not quite sure about, haven't messed with it today, but maybe something I've learned from the Child Trait will give me a necessary insight for it).
I think part of my hangup had to do with run flags and whether a sim is considered to be running or not (thanks to Eugi for pointing me in a direction where I ended up figuring that out).
What I did turned out to be fairly simple when I cut out complicated and misleading ideas. First of all, I forgot that it's Run walkstyle I want, not RunPanic. RunPanic just looks like the person is panicking, whereas Run (as opposed to Jog) looks like more of a Sprint (which is what I want).
So I ended up with code like this, ultimately, inside of a trait buff:
SKILL
27c01d95:00000000:000000002a72b9a8
RUN_ALLOWED_OUTDOORS
DEFAULT
Running indoors is already disallowed by default, as far as I'm aware, but I needed to make it so that running outdoors is disallowed, too (for the sim who has this particular trait buff). That way, they would use the Run walkstyle, no matter what, rather than trying to do Jog. Formerly, I'd been using flags like the toddler example, allowing outdoors and indoors, thinking that would somehow enable the walkstyle to work. But I think I was confusing myself on how run flags work and not taking into account that toddlers are a special case and, as far as I'm aware, don't have a default jog setting for running like most sims.
Since I'm fine with this sim basically always running, I don't think this will cause any problems when they run into priorities like fire (and they are a child, so it won't conflict with stuff like pregnancy).
I also tried using an override for run_walkstyle at one point, when I was still using the run flags differently, thinking that would override the Jog. But that was only working for what I'd roughly call "medium" distances. Long distance runs were still defaulting to Jog.
I'm not sure I totally understand what was going wrong in a way I could explain, but at least 1 out of 2 problems have a solution now!
Edit: Just realized, that's the same procedure Grim Reaper uses in its buff. I see now why it ensures the reaper walk... it's forcing the reaper not to default to jog over long distances.