Thats going to break so many websites for you ... Pretty much any service that uses server side sessions across domains. Downloads are often whitelisted to a session, which get invalidated on ip changes.
On the other hand, it might be possible to devise a solution that works generally but employs white lists or other exceptions for sites that need certain IP-address behavior. That would take a fair amount of effort, but the approach has worked well in similar contexts, such as ad blockers.