I have done exactly that.
I had a US amateur extra class license when I moved to Germany. As a legal resident of Germany, I applied for and received a reciprocal (no test required) German license with the call DJ0IQ. Yes, it was a requested/vanity call sign to match my US call. The good folks at DARC (the German equivalent to ARRL) helped me with that.
You'll need an experimental radio license (Versuchsfunk) from Bundesnetzagentur link for research operation.
For demonstration/teaching purposes, a demonstration radio license (Demonstrationsfunk für Bildungseinrichtungen) would be in order.
I'm not a lawyer nor have I ever applied for either of these; I'd recommend contacting BNetzA on these aspects. ...
As for other legal alternatives, one might be transmitting inside a suitably shielded Faraday cage. I’ve seen such facilities at labs where various prototype and pre-production (and competitors?) electronics systems were being tested to measure how far out-of legal compliance they were.
There still are some countries where proof of the CW exam is necessary to be allowed to operate on HF. So when you visit those countries and you want to operate under the CEPT-system you need to prove you did the CW exam.
In Belgium we are not required to do the CW exam anymore to get our HAREC licence but we also still have the possibility to pass it.
One other setting for each channel that may be giving you problems is Duplex/Simplex. Make certain it is set to Simplex or equivalently "no offset". This feature, when turned on, causes that channel to transmit and receive on different frequencies - typically for using the radio with a repeater.
The original reason for the CW requirement was that there was no other mode. :) The reason that I was given in 1953 is that emergency/distress calls were often in CW. Nowadays, IMO tradition-inertia is the probable reason. It would be interesting to poll German hams to see how many have ever heard a CW distress call.
I can’t comment on the specifics of the German Licence, but the wording of the terms, conditions and limitations of the UK licence is such that it sounds exactly within the terms of the amateur licence.
1(1) The Licensee shall ensure that the Radio Equipment is only used:
(a) for the purpose of self-training in radio communications, ...
In the USA, amateurs ARE allowed to use pulse modulation, which apparently includes radar pulses.
The FCC allows pulse transmissions on and above the 33 cm band.
I found nothing in Part 97 prohibiting radar.