From having played with the remote desktop offering from AWS, it’s a Windows Server running a terminal session. It’s likely heavily locked down and on its own network with likely no inbound network connectivity.
Similarly, the compute nodes are likely to be locked down to only accept connections from the remote desktop network.
It all depends on what the brief was to whomever set it up.
You might be able to do some shenanigans with the web browser on the remote desktop, but for my money, I’d just open your browser, set it to full screen and forget about how your keystrokes are travelling.
Ultimately, unless you’re a shareholder, it’s their money.
And for the record, it might be that the IT department doesn’t want you to run your own SSH session for a bunch of very good reasons.
Except that the idea is that you cannot get data in or out of the corporate network. Depending on how it’s implemented will determine how successful that is.
Regardless, you’re likely to lose your job if it’s detected without written permission and even then it’s likely to turn into a security pissing match.