what size carb for a 416 stroker

Burn your eyes exhaust is in the tune.
Your throttle blades are possibly/probably too far closed, and your idle trimmers to far open. Your engine is wanting more Idle-bypass air.

I don't know that for sure cuz I don't know anything about your build. But every single time, without exception, when I run into burning eyes, it's because the timing is too far advanced, the throttle has been closed too far, to keep the idle speed down, and sometimes the PCV is not there.

Forget idle tuning with a vacuum gauge, it doesn't work on a big-cammed V8. Know this, your stroker, at idle, will like achitload of timing; that don't make it right. It might like 20,30,35, even 40 or more degrees. And it shows it by idling faster and faster. So what do you do? You close the throttle...... which simultaneously closes BOTH air and fuel. So the engine wants to stall. What do you do? You crank out the mixture screws, and when that stops working, you say to yourself what? You say your carb is too small it can't supply enough idle fuel!, Am I right?
But the real culprit is the too much advance and the subsequent too small throttle opening, which has shut off your primary low-speed fuel circuit,namely the transfer-slots. Get your transfer slots back on line, and let the idle timing be what it will be. Then adjust your rate of advance to come in as fast as possible, without detonating, by your stall speed. Your engine does not care about power-timing below stall speed, because it never has to pull hard down there, so forget about the numbers. Just find out how much timing she wants by stall-speed, and give it to her. If your stall speed is below 3600, then your power-timing with aluminum heads is gonna want to be 32/34 degrees by 3200/3600. As a streeter it does not have to be perfect from the get-go, because it's a flipping stroker which annihilates the tires in first gear no matter how crappy slow the timing is.
Say your stall-speed is 2800. And say by trial and error you determine that 30* by 2800 produces detonation, but 28* does not. So now you have a data point. Your second data point is 32/34 by 3200, so you crank it up to 34* and that does not detonate after 3600, but does at 3200, What do you do? Well you can't run the 34 until it no longer detonates, so continue testing and the magic silence comes at 32* at 3400 so there is your next data point.
Say your idle timing ends up not stinking at 800rpm with 16* so that is another data point. Plot the points on graph-paper with rpm across the bottom from left to right, and degrees of advance vertically up the left side. Now, subtract two degrees from the higher two points, and plot the numbers on the graph, and connect the dots Finally build your D to match the graph as closely as possible.
This gives you a 2-degree safety net for final tuning.
Now go out and get your fueling and heat-range dialed in.
Then revisit the timing.
Btw, as a streeter you are gonna hate that 5.0PV. It won't open until the gas pedal is on the floor. That means you have to run a bigger than necessary MJ so you have some driveability. Cuz most of the time you will not be at WOT,lol. So what you really need to do is engage the PV earlier, so you can downsize the MJ.
My little 750DP/230* cammed 367 likes a 10.5. This makes for a smooth power-delivery, with less reliance on the pumpshot. I don't know what your stroker might need, I haven't tuned one, but I suspect it will like it. This is one place you can use your new vacuum gauge........ if you want to. Just plumb it to manifold vacuum, and stick on the windshield where you can see it. Then off for a roadtest. Get her into second gear at above stall. Then slam the pedal down to about 8" vacuum and watch the needle. What you want to see is a smoothly rising needle keeping pace with the tach, with no flat spot, or hesitation, or stumble. The needle must never go backwards or stop rising until maybe 18 to 22 inches; watch your speed. This is not a power test, rather driveability. So try it at various speeds and pedal applications, always dropping the vacuum to slightly below the 10.5 cut-in point. Also try just tipping into it right on the 10.5. If you have an AFR gauge, let it help you. But I find that the no-stumble/no hesitation rule works just as well. Do this AFTER the timing curve has been bugged out. I go to 8 inches so that by 10 the pumpshot has been used up. Watch your speed.
After you get the PV on line, you may be surprised to find that you can take a lot of pump-shot out.
And we haven't even talked about Vacuum-advance.

Almost any carb can be tuned to idle on almost any V8 P-car engine, including a SuperSix tiny 2bbl, to an 850, the biggest I have tuned.
If your idle trimmers are opened to more than 50% of their working range, chances are very good that the throttle blades are too far closed.