You have two nested ssh commands with nested here-documents, and to delay interpretation of the $ expressions in the inner one, you need more escapes. To see the problem, you can replace the ssh command with cat to see what would be sent to the remote computer. Here's an example, using your original code (and some modified variable definitions); note that the $ and > are prompts from my shell.
$ targetCompute=remote-server2
$ local1="local-srv"; echo "${local1}"
local-srv
$ cat <<EOF
> remote1=\$(echo "remote-srv1"); echo "\${remote1}"
> ssh -t -q -oStrictHostKeyChecking=no "${targetCompute}" "bash -s" <<EOF2
> remote2=\$(echo "remote-srv2"); echo "\${remote2}"
> EOF2
> EOF
remote1=$(echo "remote-srv1"); echo "${remote1}"
ssh -t -q -oStrictHostKeyChecking=no "remote-server2" "bash -s" <<EOF2
remote2=$(echo "remote-srv2"); echo "${remote2}"
EOF2
Notice that the lines relating to remote1 and remote2 have both had their escapes removed, so they're both going to have their $ expressions expanded on remote-srv1. That's what you want for the remote1 line, but to delay interpretation of the remote2 line you have to add another escape... and that escape itself needs to be escaped, so there'll actually be three escapes before each $:
$ cat <<EOF
> remote1=\$(echo "remote-srv1"); echo "\${remote1}"
> ssh -t -q -oStrictHostKeyChecking=no "${targetCompute}" "bash -s" <<EOF2
> remote2=\\\$(echo "remote-srv2"); echo "\\\${remote2}"
> EOF2
> EOF
remote1=$(echo "remote-srv1"); echo "${remote1}"
ssh -t -q -oStrictHostKeyChecking=no "remote-server2" "bash -s" <<EOF2
remote2=\$(echo "remote-srv2"); echo "\${remote2}"
EOF2
So \\\$(echo "remote-srv2") and "\\\${remote2}" in the local here-document become \$(echo "remote-srv2") and "\${remote2}" in the here-document on remote-srv1, and then the command actually gets executed and the variable expanded on remote-srv2.