The best/only way to convince management that you're both understaffed and that they need to do something about it is to stop enabling them to get away with overworking the staff they have. Really, what you need to do is stop killing yourself to get the job done, because you're only hurting yourself by doing so. You've found a way to get the job done with a third the staff that you "should" have, and because of that, management sees no need to add extra staff.
You say that you guys
have been able to put a shoulder to the wheel and get things done, so of course management isn't going to feel the need to add extra staff. The staff they have is getting it done. And by powering through and over-working yourself, you are enabling management to save money and not hire extra staff.
That changes really fast when things stop getting done, because that creates a cost for management. Right now, it's all upside for them - they're saving a lot of money, but the work that needs to be done is still getting done somehow. If you stop artificially insulating management from the costs of being understaffed, there is a chance they'll do something about it. If not, they won't, because they really have no reason to.
For what it's worth, this situation is not sustainable, and will come to a head one way or another.
There are three basic ways this happens.
The people they have will find new jobs, and you'll be so understaffed that no amount of "shoulder to the wheel" will get the job get done.
You'll all burnout, and productivity will plummet, so the job won't get done.
You guys will see the light, stop enabling this behavior and restrict yourselves to reasonable workloads.
That last one is probably the best approach for all involved, so that's what I'd advise you do - dial back on the extra work so that management starts to feel the consequences of being understaffed and has time to make the needed adjustments, because the other options create a much more rapid change in your department's ability to do work, and that tends to go badly for everyone involved - management and the employees.