There are different things:
Weight:
3,5t is the maximum for cars and light transporters. This is what one can normally drive with a common driving license. The trailer hitch is a "Kugelkopfkupplung" -> https://de.wikipedia.org/wiki/Anhängerkupplung. (All other german words in my post are mentioned in this article, too.)
Above you need another trailer hitch: Maulkupplung and either a truck or tracktor. Either with compressed air-brake or rather slow (generalized, there are exceptions). I personally plan to go for a max 18t Wechselbrücke. That means that I depend on service for transport.
Only with theoretical exceptions (aka: not seen up to now in real life) the following holds: Only 3,5 trailers are low enough for a loft.
Load:
To the best of my knowledge, you either transport with TÜV or with a red license plate. The later is generally not for individuals and you either need a service or have to know people who have one.
Either the TH gets TÜV or the TH is load and only the trailer needs TÜV. Load means, that it can be removed from the trailer without tools. E.g. by kind of container locks. Some say it is almost impossible to get TÜV für a TH, other say that they build TH on order with TÜV. I don't know the details. There are further advances and disadvances for TÜV and load but I would consider them negligible. I plan to go for load via Wechselbrücke on a Wechselfahrgestell. You might search for these in mobile.de -> LKW -> Anhänger and then select each of them as type.
An alternative to locks is a low-loading truck to transport your TH as load. Some do it even for trailer without TÜV (e.g. "Bauwagen"). Have the total height in mind!
Size:
There are two realities: The first is max width is 2,55m and max length is the total length of drawing vehicle and trailer. Typically max trailer is length 10m with the exception of 12m for Sattelkupplung. Max height is 4m.
The next reality is oversized transport. This is daily business, e.g. for yacht owners carrying their boat with a private car through Europe several times a year. Up to 3m, the minimal cost is from something around 40€ per trip or < 300€ per year. Most freight forwarders (e.g. necessary for > 3,5t anyway) will offer this service. More than 3m width and 4m height might be more expensive. This information is only based on my googeling. I personally plan to some cm width more than 2,55m. BTW: some meters additional length are no issue. You just need a warning flag in case.
Most THoW in Germany are < 3,5t. I believe this to have a psychological reason (feeling free when you can transport without service). Don't get me wrong, there is nothing wrong about it. I believe that most of our life style is not rational but based on feelings. As it is for me.