I don't see any reason why they should use a Tegra K1 or Tegra 4. Both of those use different CPU and GPU architectures and neither will be powerful enough to run Switch games as they are without any porting (and that would constrain many of the games already struggling on the X1 further, not to mention it would still require developers to put an effort); moreover, the Tegra 4 has been out of production for years and the Tegra K1 never had the volume production that the X1 enjoyed so it might cost even more to put it back into production. I think Nintendo can fit a regular X1 with thermals designed only around the handheld mode into something slightly thicker than a 3DS XL in order to fit a fan (think of something like the GPD Win series which have a fan but are overall not too much bigger than the 3DS XL).
With that said, I think Nintendo should create both a cheaper/smaller portable-only version of the Switch (only meant for handheld play) and a TV-only microconsole version (that runs all games in docked mode) alongside the regular Switch (which obviously runs games in both handheld and docked mode). Those with the portable-only version and TV-only version (think of a family with multiple children that have a family Switch connected to the TV along with a few portable-only Switch's for individual family members) can still "switch" modes through the use of physical game cartridges, a shared-digital games library (I know Nintendo would hate to do this; but lets say each account on a portable-only Switch Mini could share their games with a single Switch TV), and cloud saving/synchronization from one device to another (which has already been confirmed to be part of the paid online service). In this case, the portable-only Switch will appeal to those that want to purchase cheaper Switch units for individual members in their households, the TV version would appeal to families wanting local play and those that do not really care for portability and would rather have a more accessible price tag, and the regular Switch would be the more premium option targeted more towards young adults and to those that want flexibility and a bigger display (much like the 3DS XL).