I do essentially this with "terminal-window", mrxvt, "hcm" and "pypty".
terminal-window wraps mrxvt, just filling in some commandline options. mrxvt is a lightweight, multitabbed, nonunicode terminal emulator not dissimilar to xterm.
hcm is a GUI that makes it easy to run a shell (or other command) on a remote host. It can also start an mrxvt with remote ssh's without requiring the GUI if you prefer (using hcm-term).
pypty is a /usr/bin/script reimplementation that is written in Python. It is not significantly different from /usr/bin/script, except it gives a "dated files mode", that allows you to have one file per pseudo terminal per day. So if you leave a shell logged in overnight, you get one file per day - this tends to make it easier to find what you're looking for.
All this combines to give you pseudo terminal logging with great ease. Commands are run on remote hosts, but logged locally. Just start a "fancy terminal-window" (or use hcm-term), and everything you see on the screen plus control characters (but not nonechoed passwords) will be logged under ~/.hcm/logs/<year>/<month>/<day>/* .
Also, if you hit the shell button in the lower right of mrxvt, you get another ssh session into the same remote host, which is also logged locally (to a different file under ~/.hcm/logs/...). When I started making use of that feature, I had no idea how much I would grow to like it.
You can obtain them from http://stromberg.dnsalias.org/~strombrg/hcm/ There's a video there that shows how easy it is to set up and get started with.
BTW, fancy terminal-window sets up $PS0 or "trap DEBUG" to give you command start times and finish times. It's great for post mortems. It does this without replacing any of the usual bash startup files.
I wrote terminal-window, hcm and pypty, but I can't take credit for mrxvt. :)
HTH