load balancing over single NIC to overcome per-connection limit

Solution 1:

I don't think there's a point.

Of course, you can use multiple source IPs (whether from multiple hosts, multiple NICs or even a single NIC) but then again, you'd likely NAT all of those to the public IPv4 address you've got - so there's no difference from the outside. You can just as well just use multiple threads.

But obviously, you can just try binding multiple source IP addresses to a single NIC, not much to it.

Solution 2:

There's two ways this has been limited - on a per-VM basis or on a per-vNIC-basis. If it's on a per-VM basis there's nothing you can do other than threaten to move to a different provider or actually move providers, obviously the former might not work. If it's on a per-vNIC basis then you could either add more yourself if that's possible or again ask the provider to add them, and again if they refuse threaten to leave.

Ultimately you have a business need that it sounds like isn't being met by your provider, so be prepared to pay a little more to get what you need.