They make special pens for this. https://www.amazon.com/Mental-Health-Non-Lethal-Flexible-Point/dp/B00KALP5Z4
The other obvious workaround, if the facility allows, is to lend them a pen to sign something with and then take it back. One of the doctors I worked with did have someone disassemble the pen and steal the ink part from it during this process, which the doctor didn't notice until afterward, but he can't have been paying super close attention.
Well, the fundamental tension is between stupid bureaucratic rules, not anything that's easily engineered around. You might start by looking for the definitions of "ballpoint pen" in the requirements, and the specific wording of the prohibitions of the prisons/hospitals involved. You may find a loophole where you can construct or find an implement that satisfies both.
My guess is that they're under-defined in text, and up to the judgement of the bureaucrats or guards. So bribery may be a solution. Or lobbying to change either side of the rules (like ability to use a pen while supervised).
If this is worth putting some effort into, a mix of lobbying and identifying a relatively-hard-to-abuse and still permanent ink device would be the direction I'd take. I don't know if it'd be more successful than simpler lobbying to relax either or both rules.
As I understand it, the rationale is a) you can stab people with pens, b) if you snap the barrel of the pen in half you have some sharp edges to play with, and c) The Rules Say So. If that's the case, you can get around all three of those problems by removing the ink chamber and the ballpoint tip from the rest of the pen, and letting them write with that (I tried this just now with a cheap Bic pen, and the only problem I had was my handwriting suffering a little).
Maybe have them handle the pen though a glove box?
Ridiculous, yes, but possibly the kind of ridiculous that happens in real life.
In Scott Alexander's article about his IRB nightmare, he writes that one of the problems he faced is that the rules both said that documents have to be signed by ballpoint pen and that the involved patients are only allowed to use pencils.
Politically, there seems to be a similar issue in the case of Joshua Schulte who's imprisoned under poor conditions by the US and who's complained were rejected because they were not filled out with a ballpoint pen when he was also not being able to use ballpoint pens due to being under suicide watch rules.
Is there any way to construct a pencil to fulfill the legal criteria that would allow its use in those cases?