No IPv4 on LXC

You must mean public IPs from your ISP?

If you mean local IPs then that's nearly impossible.

You can just change your /24 subnet [255.255.255.0] to a /23 subnet [255.255.254.0] which would double your available IPs... All the way to a /16 subnet which would allow 65534 available IPs.

For example 192.168.1.0/23
Available IPs: 192.168.0.1 - 192.168.1.254
 
So all you should need to do is configure your host with an IPv6 address.
Configure the container network interface for IPv6 DHCP or static.
 
Reverse proxy sounds interesting. I have to look in to that, maybe you can point me in the right direction, but I'm still wondering if it's possible to run a LXC container without IPv4?

The reverse proxy will act as a sort of gate where all the queries will enter there and be redirected to the appropriate backend server. It means you will only need a single public WAN IP for your proxy and for the backend servers you will use a LAN IP. Way more scalable and easier to manage. The main 2 are Nginx and HAproxy. Here's an article which explain some benefits of reverse proxies.

I cannot comment on IPV6 for LXC since I'm not using it.