Which external flash works best for Nikon D850 wildlife photography?

Does anyone have a solid recommendation for an external flash that can actually keep up with a D850 for high-speed wildlife stuff? Ive been shooting with the D850 basically since it launched and usually I just lean on the ISO because that sensor is such a beast but lately Ive been doing a lot more work in the heavy timber near my place in Oregon and the light is just non-existent under the canopy. I tried digging out an old SB-700 I had in the gear closet last weekend but the recycle time was absolutely killing me when I was trying to catch a woodpecker moving between shadows and I missed like half the sequence because the flash couldnt keep up with the burst rate.

I'm looking for something with serious reach because obviously I'm mostly using the 500mm PF or sometimes the 200-500mm zoom so a Better Beamer is definitely gonna be part of the kit but I need the flash itself to be smart enough with TTL so I'm not blowing out the highlights when the light shifts. I was looking at the SB-5000 because of the radio control and the cooling system which seems like it would help with the rapid firing but I dont really do multi-light setups in the woods so maybe its just expensive overkill?

Then there is the Godox stuff like the V1 or the V860III with those lithium batteries which people swear by for the recycle speeds but I've heard some mixed things about Godox and Nikon TTL accuracy when youre shooting through thick brush. I really cant be fiddling with menus or manual power adjustments while a Great Gray Owl is staring at me from forty feet away.

Budget is around 500 or 600 bucks but I could go higher if it actually makes a measurable difference in my hit rate. I have a trip out to the Olympic Peninsula coming up in three weeks so I really need to get this figured out and spend some time practicing in the backyard first. Is the SB-5000 still the way to go for the D850 or is there something better for wildlife specific needs nowadays...