I subscribe to Apple One family - what happens if I buy AppleTV 4k and get 3 months free AppleTV+?

The offers are intended for new subscribers, and will automatically allow redemption for those users when setting up their new device. As above, you would need to contact Apple support to see about any exception. Just be aware that there will be no guarantee.

it will be lost unless you cancel the one family I believe

if you wish to have it verified by apple then you will have to use their support

Apple Support