For what it's worth, I'm in the USA (where I do suspect this originated) and while I wouldn't say it myself I've heard "get with/got with" in several different contexts.
It's informal (colloquial, slang) and would be completely inappropriate to use in a court document here, unless as a direct quote, because it's vague and unspecific and HAS no established meaning and therefore is likely to be misinterpreted unless used it a very clear context. Could you challenge it on that basis?
Some examples of uses:
(1) A sexual encounter, or something physical short of sex (kissing, touching, etc.) Not necessarily negative. "I want to get with Jake at the Spring Dance - help me pick out a sexy outfit."
(2) The start of some kind of relationship: "I first got with Jake at the Spring Dance, and we've been together ever since" doesn't necessarily mean we had sex then, more that that's when our relationship as a couple started. But if someone else says "Wackamole got with Jake at the Spring Dance" - yeah, it means sex or something approaching it.
(3) A sexual encounter (could be repeated, or a full-blown relationship) with someone who is already in a relationship: "I can't believe Jake got with Katie while his wife was pregnant!"
(4) It COULD be nonsexual - for example in a work context: "get with Jake in AV and see if he has any visuals we can use for this marketing campaign..." However you would absolutely NOT say "I got with Jake" without qualifying the context, because people would assume it was sexual.
If a new friend said "I got with Jake back in college..." I would have no idea if she meant that she had had a one night stand with Jake or that she'd had a long, serious relationship with Jake, or something in between.
I'll also say "she got with a man" sounds like a very odd thing to record - if it's relevant, why not specify the man's name? This is the part - along with the lack of context - that gives the negative/shaming impression and implies something exclusively sexual.