I have the same mouse and the issue isn't with the mouse but instead with the Corsair iCUE software.
I've had that issue randomly for the past 3 or so years and in that time, i've also replaced my mouse under warranty (the middle mouse button stopped working at some point).
Of course, i also have a plethora of Corsair peripherals as well, that my iCUE has to manage (KB, mouse pad, headset stand and headset - the full suite). So, randomly, some of my Corsair peripherals doesn't start working when booting up. I've had to replug my KB, my mouse and my headset for them to start working. I've also went through several iCUE versions, as iCUE is getting updated and i still have that issue.
Moreover, on my 2nd PC (Haswell, full specs with pics in my sig), i also have Corsair peripherals, which are almost the same ones as on my main PC (Skylake, full specs with pics in my sig) and even on there, i've had issues of Corsair peripherals not starting on boot up at rare instances.
Here, replacing your mouse doesn't fix the issue. Though, you can go through the trouble if you like. However, you can create a support ticket with Corsair to get more help from them.
Though, i haven't done it so far since i can not reliably recreate the issue and it's completely random at which boot up my peripherals do work and on which ones they don't. So, me complaining to Corsair about it, without being able to give Corsair solid data, so they can reproduce the issue on their end, isn't helping. At this point, i've gotten used to it and can live with it comfortably, while hoping that at one time, new iCUE version fixes it once and for all.
However, if the issue is annoying to you, feel free to contact Corsair about it (i doubt that they can help you). But at least, it's something.