I recently replaced my stock dunlop AT20 tires with Falken wildpeak a/t3w. Same size.

I guess it's not a complete perfect science, but should the falken be running at the same pressure as the stock tires? Or should the "default pressure" but something different for these tires?

Stock pressure should be at least in the ball park but what I do is get some chalk and find a flat parking lot or someplace else open and as flat as possible. Put a line across the tread of each tire and drive straight a few feet. If the wear is even then you are there. If there's more chalk wear on the outside edge, it's under inflated, more wear in the middle it's over inflated. Adjust pressure with a portable air compressor and you have the correct pressure for your vehicle with those tires. A little time consuming but you know you have it right.

Same size, but are they the same load range?

If they are P-metric, then recommended stock inflation pressure applies.

If they are LT tires, they can require more pressure to match the OEM tires load carrying ability.

However, P-metric tires have their load capacity reduced by a factor of 1.1 (according to the FMVSS), when P-metric tires are mounted on light trucks or SUV's.
So that needs to be taken into account when mounting LT tires on a vehicle which came with P-metric tires.

Another factor is that the recommended inflation pressures are sufficient for the max. GVWR of the vehicle. My 4Runner will never be at it's max. GVWR, so technically I can get away with lower pressures than are necessary to accommodate GVWR.

Sub'd

Falken lists a P metric tire in the stock size and an LT. P metric would be the same pressure as stock, the LT tire would be higher pressure.

Okay good to know

Mine is the P metric...

I have the falkens. I run about 39lbs in mine. I think 36 is the sweet spot but I run a little more pressure to try to get a little better mpg and I am use to a firm ride so it doesn’t bother me.

My year+ with the wildpeaks I run them 38 on road 28 off-road and they have performed perfectly BUT shimmying has been an issue I finally have them to an acceptable shake which disappears as they warm up
Mike

What do you mean shimmying? I feel like i might be having some issue but not sure if it's a balance or alignment issue or something else

I have the same tires (P-metric) in 285 and the shop set them at 38. I'm curious what the lower limit is that I can lower the pressure on these tires off-road.

The Falken Wildpeak A/T3W's have a history of uniformity issues. Meaning that far too often they are not round, and frequently can't be balanced so they ride smooth. It was my main reason for not going with a set.

You might get lucky and get a round set, or a competent tech that truly knows how to use a RoadForce balancer to it's full capability, might be able to get a set to tolerable smoothness levels.

The 4Runner has a well-made steering system, with little to no slop. The downside is that any imbalance or tire uniformity issue is more likely to be noticed.

The shimmy is NOT falken related
For winter I mounted new blizzaks and have shimmy that goes away as the tires warm up
4runners are extremely sensitive to tire pressures and balance
For the mild off roading I have done 25 psi was as comfortable as I would go but others have aired down here on the list to 17
On road 38 is what I run I’ve seen here anywhere from 33 to 40 it depends on how you like the ride or what suspension you have
Mike

I worry I MIGHT have this problem, but have already driven 1,000+km on them... what should be my next steps?

Nothing if you have not felt anything

If you're feeling a vibration or shimmy on a new vehicle, it is almost always the tires, and/or the balancing job. I'd go back to where you bought the tires, and if they have a RoadForce Balancing machine, have them re-balance the tires utilizing the full capability of the machine. And a technician that cares enough to take the effort to optimize the tire-wheel assembly.

If that doesn't do it, have them warranty the defective tire(s) and bring in as many as necessary until they find tires with good uniformity.

Good luck.