You are right.You need to point original site to some proxy (for example, nginx) — it is done by A record, so RouterOS can do this for a long time now; then redirect user's browser to mirror site. PROFIT.
My server has authoritative=yes, but still getting of an addres for WiFi client is slow.Enable authoritative in DHCP server:
/ip dhcp-server set <name> authoritative=yes
I changed to 192.168.1.0/24, but NAT Loopback still doesn't work.That may or may not be part of the problem, but it would be the first thing I would correct. Your internal subnet should be somewhere in either 192.168.0.0/16, 172.16.0.0/12, or 10.0.0.0/8 space.