More info is needed, really.
Yes you coudl ssh there and use the local Emacs (v23 apparently.)
If ssh is supported like that, then local to you Emacs (v 29 or whatever) over Tramp/ssh should work fine as well (the F5 load balancer deep packet inspection wouldn't see the different; Tramp is just ssh'ing and running commands same as a user would.)
You could also try sshfs (map a local filesystem to a remote mount over ssh) but in my experience thats really slow.
You could run an Emacs on the remote, with X11 projection to your laptop, over ssh tunnel, but that'd likely be slow, and Emacs 23 is ancient :)
.. so a local to you Emacs 29 with Tramp over ssh to the remote should likely work, but it depends on some other things; they _could_ block it, but not likely, assuming you can ssh there yourself.
Now, if theres a jumpserver or something, that could be a problem (2FA type challenges in the middle) but multi-hop ssh tramp could do it..
Another question is .. why the heck are they stuck at Emacs 23, which is probably 10+ years old. Thats a security risk right there they should be happy to remedy :)
Is it perhaps the wrong TERM setting, so you're using say xterm TERM but its actually being run in text with ANSI mode or something? Which ssh client are you using?
Goodness; that version of Emacs was released in 2013, a full 10 years ago :O
Can you not get that updated to something remotely more recent? :) I'm surprised you don't run into a lot of issues, if you're using remotely modern packages.
If you run a remote LSP and connect to it from lsp-mode, can the file saves be sent through that? Or does thr LSP only do file checks and refactors and such, not offer raw file get/put?
Wow quite a journey! I learnt Emacs decades ago so am wired to the usual key combinations, but I feel like devil-mode will be useful as I age, or if injuries make it harder to use two hands at once.
As its brand new, I'm ignoring it for now; when I don't have another more mature ("the devil you know") mode, I'll use the *-ts-mode and twiddle with it, but otherwise I'm mostly sticking to the current ones. One of those bleeding edge things - let those on the edge do the bleeding ;)
Examples of various commands? Screenshots of what you're referring to?