I am convinced that an installation without a picture would work with Linux on the AsRock. But you must be a profound connoisseur. But then you wouldn't ask the question like that. Simply changing the CPU after the installation would also be a possibility (as Me described).
But let's be direct (I have 25 years of Linux experience); what if problems should occur during booting? what do you miss first?
The power consumption drops immediately if you don't have a screen plugged in, or are you concerned about the number of CPU cores and the CPU cache? My personal recommendation would be to go for the 3400G.

You can't install Linux without an image either - but after that you could also use a 1600 with SSH. Of course, 3400G works best with a picture.