Update the bios with a character remapping Razer Tartarus V2?

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It doesn’t have onboard memory, so you have to have Synapse running in background.

You are correct in that it does not have any onboard memory. You do not, however, have to have Synapse running in the background. So it does have a “default” setting. This default setting is what I believe the OP is asking to change, since he is referring to “the bios”. Personally I do not think it is possible, but it would indeed have been good if it were. As Synapse 3 is a truly horrible piece of software, and the Tartarus actually works far better without Synapse than it does with Synapse. I have had soo many issue with the Tartarus and Synapse, all of which have been directly proven to be in Synapse itself, not that actual Device. So I stopped using it. If you use it for work, Rewasd is a good option, as it works far better than synapse, however quite a few games have banned it since it can be abused to gain advantages in gaming.