For the purposes of this question, let's simplify what VPN operators do as "they allow their customers' internet protocol (aka IP) addresses to appear as other (veneer) IPs to the counter-parties to which those customers connect."
I am fully aware that there are private IPs that are mapped to public IPs for the purposes of routing connections. This is not what's at question here. At question here is the arrangement in which public IPs are masked by presenting traffic to site operators as appearing to originate from the veneer IPs of the VPN operators.
This allows (for example) users in a country A to route their traffic through a commercial VPN operator in a country B and access services in country B which are restricted to users from country A. This can be used to subvert both intellectual property restrictions on traffic and legal restrictions that site operators attempt to enforce by geography-based restrictions.
At first glance, it may seem that it would not be impossible to track all veneer IPs and identify them as such publicly. So a knee-jerk response to this question might be "there is no need for such a law." However, while it may be possible to track all IPs used by a commercial operator of a VPN which engages exclusively in VPN business, it is not possible to track all IPs dedicated to VPN business by an operator which has many other forms of business.
So, for example, NordVPN's IPs can be identified as belonging to NordVPN, but IBM's or Google's IPs cannot be identified as used exclusively in their VPN business.
Since there is a public policy interest (mentioned in the 3rd paragraph of this question) in knowing which IPs are VPN veneer IPs, Congress may decide to require commercial VPN operators to register those veneer IPs in a way which would make them clearly distinguishable from non-veneer IPs used by the same entities.
Would that be constitutional? More specifically, is there any case law which may be relevant to deciding whether this would be constitutional?