If your computer initially arrive with Windows 8 and if it was later upgrade to Windows 10 [note upgraded from 8 to 10, not clean installation of 10], then you no longer need product keys. Your computer's Win 10 lic is activated using new activation method called Digital Entitlement. Digital entitlement will automatically activate your PC as long as hardware configuration is the same.
I cannot comment on behalf of Microsoft but to my knowledge and what it used to be, SSD/HDD change is not usually considered huge change. I am not sure if Microsoft support understood you well, but SSD change is not considered huge change - it happens on daily bases and there are hundreds of people who replace faulty HDD/SSD with new ones and reinstall Windows. However, you may need to verify this with MS support.
More info >> //support.microsoft.com/en-us/help/12440/windows-10-activation
\>> //www.thewindowsclub.com/digital-entitlement-activation-methods-windows-10
\>> //www.microsoft.com/en-us/software-download/windows10ISO
You can now download an ISO image from Microsoft web site [make sure you download the correct version/edition] , use it to reinstall Windows, do not enter any product keys and eventually it should activate itself without codes because it will detect the same product, same motherboard, same serial number, same hardware configure despite bigger SSD.
There is a new feature in Windows 10 coming with the latest Anniversary update (do you run with Anniversary update?) that can help you link your Win lic with your Microsoft account. Therefore, if you do it and next time after reinstallation with new SSD it will probably help you verify its you . More info >> //www.windowscentral.com/how-re-activate-windows-10-after-hardware-change