How do I minimize the number of passwords leaked when a PC gets compromized?

For customer support reasons, we need to store passwords to some of our customers’ systems (with their explicit, written permission, of course), as well as, obviously, passwords to some of our own systems. Customer support agents and administrators need to be able to access those passwords.

I am aware that – despite all our efforts to the contrary – it might only be a matter of time before one of the PCs in my company’s network gets infected with malware.¹ When that happens, I want to minimize the damage. In particular, I want to minimize the number of passwords that gets leaked.

Classic password managers don’t help here. They help against password reuse and other dangers, but they are not designed to mitigate that kind of threat. On the contrary: The customer support guy’s PC is compromised, they enter the master password to their password manager and… bang… the bad guys have won the jackpot.

I am looking for a solution to ensure that only as few passwords as possible (ideally, only the passwords actively used during the time period between the PC being compromised and the attack being detected) are leaked.

Obviously, storing the passwords on paper² instead of a password manager would solve this, but I would also like for the solution to be practical (the paper solution won’t work if the users are in pandemic-induced home office, for example). Another option might be an online password manager which "rate limits" the number of passwords that can be accessed per hour (and sends out e-mail alerts when too many password are requested).

I don’t want to reinvent the wheel, and I’m sure that other companies have the same issue. Is there any "canonical" or at least widely-used and established solution to this issue?

¹ If you google for "percentage of companies having been hacked" or "percentage of companies having been infected with malware", you get various news articles on the first page claiming various two-digit percentage rates. I don’t want to get hung up on a particular number, I just want want to illustrate that the risk is real and much, much more likely than "rubber hosing" or something similar.

² To clarify the threat model: I am concerned about hackers on the other side of the world having a "lucky day" (i.e., an employee starting a malicious file which somehow managed to get through all our filters), getting into our network and then seeing what they can harvest before doing their usual ransomware stuff (yes, we do have backups). I am not concerned about targeted and/or physical attacks by state actors (fortunately, we are not important enough, and neither are our customers).