Yes, you will lose a full stop with the teleconverter. Let's break this down and compare it with another option.
Current Nikon prices, new Note: some of these items are on sale.
Z100-400 F4/5.6 $2700 1.4 x teleconverter $590 Total = $3290 Max focal length 560mm @ F8 total weight 58.4 oz
Z 180-600 F5.6/6.3 $2100 Total = $2100 Max focal length 600mm @ F6.3 total weight 68.96 oz
To me it's very clear that the 180-600 is the logical choice IMO. The 100-400/1.4 TC combo is more expensive, has smaller max aperture (which will likely affect AF), is external zoom, and may not be as sharp due to using a teleconverter. That last item is supposition on my part due to the reputation of TC use, especially with zoom lenes. The downsides to the 180-600 are that it weighs more, and you are limited to a minimum focal length of 180 vs 100.
I do a fair amount of bird photography. I fell that 600mm is a great focal length for most of my bird photography. I currently own the 180-600 and use it with a Z8. I really like this lens. It is very sharp and handles well. It replaced a 200-500 F mount lens using an FTZ to mount to the Z8. I particularly enjoy the internal zoom of the 180-600. I no longer worry about dust and moisture creeping into the internals every time I zoom. Also external zooms tend to sag with age and use, which I'm sure affects performance.