In general, for proxies all that matters is cores and network bandwidth. Velocity can use all your CPU cores and pushes a lot of traffic.
The primary CPU cost of the proxy is roughly split between Velocity-side processing (such as compressing/encrypting packets and running plugin tasks) and the operating system kernel (for I/O). Expenses in one side affect the other as well, since they compete for the same CPU resources.
Onto your setup: With the i7-7700K you’re getting 4 cores, which is respectable but not ideal. An optimized proxy setup aimed at hundreds of players online would focus on CPU cores, even if they are relatively slow (8 2.3GHz cores are better than 4 4GHz cores). Nonetheless, the i7-7700K should still provide good performance, but it is less ideal, so be careful.
Seeing 266 players per proxy bogging you down on BungeeCord is pretty bad. Velocity should be able to manage more than that (even in a relatively low-power cloud environment like my network is on, we’ve pushed 300 players on a single proxy with room to spare). Unfortunately, again, I am unable to provide hard numbers since everyone has a different setup, but your 3 boxes should be able to handle Velocity with 266 players just fine. If not, I’m very interested in fixing the problem.