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So I had this dumb idea to mount an enormous pewter eagle model way up high over its base to make it look like it's launching itself off a rocky peak. I can get it so the center of balance is within the center of the base easily, so when it's just sitting there it's fine, but any nudge is going to send it flying and messing up the paint I haven't put on yet. Anyone have any ideas on how to stabilize it? It's just a lone 40mm square base, so I can't really magnetize it down to a big tray or anything.

Those can be a bit tricky.

The simplest answer of course is just to make the base larger...that isn't always an option.

With it being pewter (I assume lead free?) - you can do what I do a lot and make the base out of a solid chunk of lead.

The lead weighs about twice as much as lead free white metals (depending on specific alloys and what not...). You basically get yourself a lead block, cut it to size, carve it as desired and then mount your miniature to that.

Of course there can be concerns working with lead - but they are minimal unless you are doing silly things. It cuts very easily, sculpts with pretty much any tool that is harder than it (I normally just use a set of dental tools). Don't eat it, smoke it or lick it. Wash your hands when you are done.

Steel/iron washers can help some - but your really need to get the CG down to around 20-25mm above the play surface for it to be stable on a 40mm base.

Glue into the bottom lead shot.
Small rare earth magnets do fit in those little round bits you know:

Add scenery on the base that involves rocks or carve up some model train weights to look like scenery to be glued onto the top of the base.

I think a combination of techniques will be in order.
Weighing the base down or recasting it in metal or adding rocks. Privateer press make metal ds style bases but I've never seen square ones?
Would it be possible to drill out some material from the eagle and re sculpt/fill where you've removed material from?

Adding Heavy Rocks to the base is always nice if you can model it to blend together with the base (always a fun project)

Filling with lead shot works,
making a metal base works too. if you REALLY want this thing to not fall over and have access or dont mind paying you could always get a nice chunk of steel milled into the right sized base and use a brass or steel rod to mount the thing. but thats just cranking it to 11

I like to get some metal washers from Home Depot and glue them into the recessed area under the base.

Just went through this with an Astorath conversion

I ended up having to remodel it a few times. What I did was made it so the wings were closer together, creating a more balanced center of gravity. Even so, it was still easy to tip. I then glued a nickel into the base and filled it with putty. Then I added some rocks and metal bits to balance it further.

Washers, bolts, or such can be a cheap way to add weight to a base at minimal cost and no real helath risks.
Depends how much you need though to baslence it.

Where would one find a chunk of raw lead? o.O

Will definitely be adding in some extra-heavy scenery at the bottom, thanks. I'll be keeping a close eye on the CG as well, thanks for the benchmark measurement. Good goal to shoot for.

Hobby shops, or anyplace that sells pinewood derby stuff.

Fishing shops usually sell really cheap lead. Coins super glued to the bottom work surprisingly well too.

I have eagles for my WHFB elf army that have to rank up.
Because of base size, I have to mount the outer two off centre- and even with magnetised bases (to the tray), they still fell over.

My solution was small rocks from the garden (on the INNER sides) to counterweight them. There's enough clearance on the middle one's base for them to overlap slightly as well

I use tungsten disks. Tungsten is denser than lead so you need less of it to balance the metal of the model. Still you can need quite a bit for something like a metal Zoanthrope.

I use washers. Cheap and easy to find. Also, any place that sells pinewood derby stuff, you can get long thin strips of metal that also work great.

Large flat disk of clear plastic under the base, to give the base a lip?

Make the base rather heavy, if you got any lead, add that to the base, if not, get some metal disks and glue it to the bottom of your base