Finally I dug up a USB to SATA adapter and plugged the disk of the Ivy Bridge-E + GTX1070 into another Linux x86-64 computer (which doesn't have a dGPU). After I copied the boinc data directory, I edited the config files to suspend computation and made the GPU detection file sticky (both steps may have been superfluous), then started the client with the copied directory as data directory.
I then successfully reported a dozen tasks which had already been completed on the 1070 before the shutdown, and aborted + reported the remaining tasks. The computer ID was preserved during this transition, merely computer name, CPU type, and operating system showed up changed on the setiathome host web page, reflecting the properties of the new computer now.
So, at least at SETI@home and while using the anonymous platform, it is possible after all to fetch tasks on one physical computer and report them after completion from a different physical computer.
It took a rather long while from the reporting until the new task state was reflected at my results tables at the project web site (half an hour maybe).
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Edit,
I forgot: An important step was to switch SETI@home to "No new tasks" on the new computer, before it would needlessly request new work. Since I configured the suspension of computation before I started the client, I could conveniently switch to NNT after the start. Alternatively, I could have added <dont_request_more_work/> to the SETI project section in client_state.xml before I started the client.