The packet cloned through TEE, is set to be untracked by conntrack, prevented from being duplicated again, and emitted through the local output part of the routing stack, so first seen by raw/OUTPUT (you can also refer to this Packet flow in Netfilter and General Networking schematic: that would be right after the local process). This would be the same if the duplication was done before (eg in mangle or raw PREROUTING) for a routed packet. This makes it difficult (but not impossible) to receive additional treatment like an additional nat, since it won't be tracked and it's difficult to distinguish it from the original.
Here's an example TRACE (using iptables-legacy's version) capture in a setup similar to OP's of a ping made from 192.168.0.2 to 8.8.8.8: the capture was on the router using iptables -t raw -A OUTPUT -j TRACE. The original routed packet will not appear since it would have to be captured from raw/PREROUTING, thus validating the explanation above telling it's done from output.
TRACE: raw:OUTPUT:policy:2 IN= OUT=eth1 SRC=192.168.0.2 DST=8.8.8.8 LEN=84 TOS=0x00 PREC=0x00 TTL=63 ID=39209 DF PROTO=ICMP TYPE=8 CODE=0 ID=3070 SEQ=1 TRACE: mangle:OUTPUT:policy:1 IN= OUT=eth1 SRC=192.168.0.2 DST=8.8.8.8 LEN=84 TOS=0x00 PREC=0x00 TTL=63 ID=39209 DF PROTO=ICMP TYPE=8 CODE=0 ID=3070 SEQ=1 TRACE: mangle:POSTROUTING:policy:2 IN= OUT=eth1 SRC=192.168.0.2 DST=8.8.8.8 LEN=84 TOS=0x00 PREC=0x00 TTL=63 ID=39209 DF PROTO=ICMP TYPE=8 CODE=0 ID=3070 SEQ=1