I have solved this problem! The trick is to use two Docker containers:

1. Gluetun (https://github.com/qdm12/gluetun): set this up to connect to your VPN.
2. Tailscale (https://tailscale.com/kb/1282/docker/): set this to use the Gluetun network.

Here is an example docker-compose.yml:

version: "3"
services:
  gluetun:
    image: qmcgaw/gluetun
    container_name: gluetun
    # line above must be uncommented to allow external containers to connect.
    # See https://github.com/qdm12/gluetun-wiki/blob/main/setup/connect-a-container-to-gluetun.md#external-container-to-gluetun
    restart: unless-stopped
    cap_add:
      - NET_ADMIN
    devices:
      - /dev/net/tun:/dev/net/tun
    volumes:
      - ./gluetun:/gluetun
    environment:
      - VPN_SERVICE_PROVIDER=airvpn
      - VPN_TYPE=wireguard
      - WIREGUARD_PRIVATE_KEY=xxx
      - WIREGUARD_PRESHARED_KEY=xxx
      - WIREGUARD_ADDRESSES=xxx
      - WIREGUARD_MTU=1320
      - SERVER_COUNTRIES=United States
      # See https://github.com/qdm12/gluetun-wiki/tree/main/setup#setup
      # Timezone for accurate log times
      - TZ=America/New_York
      # Server list updater
      # See https://github.com/qdm12/gluetun-wiki/blob/main/setup/servers.md#update-the-vpn-servers-list
      - UPDATER_PERIOD=24h

  tailscale:
    container_name: tailscale
    cap_add:
      - NET_ADMIN
      - NET_RAW
    volumes:
      - ./tailscale/var/lib:/var/lib
      - ./tailscale/state:/state
      - /dev/net/tun:/dev/net/tun
    network_mode: "service:gluetun"
    restart: unless-stopped
    environment:
      - TS_HOSTNAME=airvpn-exit-node
      - TS_AUTHKEY=xxxxxxxx
      - TS_EXTRA_ARGS=--login-server=https://example.com --advertise-exit-node
      - TS_NO_LOGS_NO_SUPPORT=true
      - TS_STATE_DIR=/state
    image: tailscale/tailscale