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Turns out that LinkedIn support was able to help me.
If someone runs into this issue: special characters (_ | ( ) [ ] { } @ # * ~ < > \) need to be escaped in the JSON body, this also applies to characters in the mention name and URN. So, a mention of this type
@[Name with (special) data](urn:li:person:_123as31)
needs to be escaped to
@[Name with \\(special\\) data](urn:li:person:\\_123as31)
All spelled out in the little Text spec at the URL I mentioned above, just not that this also applies to mention names and URNs. Further caveat: \ gets escaped to \\\\ (four backslashes).