I have a fair amount of experience with the Lenz and Hotronics socks. In my experience the Lenz 6.1 and the thin Hotronics fit similarly. Both of these socks are marginally thicker than a very thin race sock. They *might* be a problem with someone who has an extreme performance fit to their boots, especially in the toes/forefoot. I ski in a Nordica Promachine with 1cm behind my heel when I do a shell fit - I did make minor adjustments in the toe area of my shells to improve my comfort, although I probably would have been fine without making any adjustments.
I have been using boot heaters since the early 80's and probably could not ski without them - my feet get cold when the ambient temperature is below 60 degrees F. I have used various brands of boot heaters over the years, and my experience is that the heated socks are just better for keeping my feet from freezing. The important feature for me is the surround feature of the heating element in the toe of the socks that I buy - not all of the Lenz and not all of the Hotronics come this way. The Lenz 6.1 is currently the thinnest sock that they make and has a surround heating element. When purchasing Hotronics socks, you have to be a little careful: they sell "Classic" socks (with heating element under the toe only) and "Surround" socks (heating element surrounds the toe); they also sell "Comfort" (thicker) and "Thin" (well, thin) in both styles. I use the "Surround, Thin" socks. In my experience, both brands run a little large - size down if there is any doubt in your mind.
Interestingly, my feet tend to sweat a lot in regular ski socks, and I do not have this experience in the heated socks (either brand) - in the past I would need to use antiperspirant on my feet to prevent sweating - this has not been an issue with the heated socks. I have decided on the Hotronics as my go-to heated sock. I have two pairs and have not had any issue wearing them multiple days in a row.
The bluetooth function that each brand comes in is healful - Lenz has an rcB battery and the Hotronics system is XLP 2P BT. Generally, the Hotronics system is cheaper than the Lenz system., and both are more expensive than standard boot heaters.