Reproducible builds are fantastic. But I doubt that, in this instance, it would be a valid way around the GPL.
Maybe if one, as an art project, made a project that, when compiled, had a particularly aesthetically pleasing binary representation, it would be okay. But we’re talking specifically about a signature needed to run the software, which appears to fit the definition of “Installation Information” in section 6 of the GPLv3.
Note that this is VERY limited in how much it could possibly help. But I don't see why it wouldn't be permissible by all licenses involved. It doesn't require the signing key, but a valid signature. For that device, it is locked down enough that the only installer that can work is the one that matches this signature. If installing somewhere else, you don't need the signature anymore, and can make your own or use an unsigned loader.
Now, it seems what you are implying is that they need a new signature that must be created at build time. I'm asking for a much more restrictive bootloader that would have to be buildable with a repeatable process so that someone else can create the same one down to the bit level. Is this as useful as one that is not as restricted? No, but it would be a way past the legal problem?