The Nintendo switch doesn't run most games at "1080p", in fact the game DOOM drops down to about 640p or so.
The 1080p output is simply whatever the game resolution is, then quickly scaled to 1080p and output via the HDMI connection when docked or 720p on the screen (or just 720p when docked I'd have to investigate that.. wouldn't make a difference).
4K and 1080p should look exactly the same within the limits of whatever difference in video processing is done to the 1080p signal, but that won't be significant likely (except to possibly add a lot of LAG due to the HDTV's processing time so you want a "gaming mode" type option for the HDMI input, or know that the HDMI input is low latency (i.e. under 40ms).
So...
Not sure why you say 1080p games look "horrible" or even what you mean by it because most can look great. Possibly a little bit of fuzziness to the HUD text but if you do anti-aliasing via MSAA, or other method most of the jagged edges get smoothed out.
Depends on the game though, as games with small text/HUD elements tend to look worse, whereas games like SKYRIM can game at lower resolutions and look pretty good.
*So games for the Nintendo Switch won't tend to have small text due to the limited pixel count and how large the screen is to your eyes in mobile.
To repeat, the 4K HDTV will not matter vs a 1080p HDTV. Either way you're sending out a 1080p signal which again is probably 720p or lower so, DOOM for example is:
620p->720p upscaled for mobile
620p->1080p (or 720p?) via dock
If you have a 4K HDTV then it is:
620p (average)->
1080p (Dock output)->
1080p-to-4K (HDTV video processor)
You are NOT meant to fill up your field of view too much due to the relatively low resolution of the Switch, so essentially it's best that your 4K HDTV look roughly the same size as the Switch does when sitting from it.
Sitting close to a MONITOR however will reveal the softness of the image. The TEXTURES are not high resolution. You're simply upscaling from the 720p or whatever resolution the actual game is, and are NOT doing upscaling then downscaling after the fact to act as some sort of anti-aliasing.
So long story short it will look:
fuzzy/blurry likely when viewed closely (just hold up the Switch closer to your eyes so it looks the same size as your 4K monitor would)
I repeat, it does not offer any kind of upscale/downscale beyond 720p AFAIK.