Can I purchase shares of a privately held company as a non-accredited, non-US citizen?

Can you perform consulting services that would be useful to the company? You could gain ownership by consulting in exchange for stock options, much like an employee is compensated in stock options.

Allowing you to invest would seem to be legally onerous for the company, whether you are considered non-accredited or simply a foreign investor.

Yeah absolutely, you can lie about being accredited. The financial jargon for this is called "self-certify" or "self-certification". Imagine that finance terms are all euphemisms cackled about over martinis at the going away party for a Goldman Sachs Vice President that just got nominated for SEC Commissioner and it makes a lot more sense.

In Regulation D 506(b) offerings, self-certification is as simple as you saying you are accredited, and not giving any indication to the contrary, such as not being able to make a minimum purchase amount, or making a post under your real name saying that you aren't an accredited investor.

In Regulation D 506(c) offerings, self-certification requires corroboration with you being able to prove you are accredited or having a lawyer say you are accredited. Here, your financial circumstance can have changed at any time, so it doesn't matter what you previously wrote.

The consequences from the government are solely on the securities seller, which is the only way accreditation laws can withstand constitutional scrutiny in the US. Buyers, like yourself, have no actual prohibition from buying anything. The state made it so that the entire financial sector and securities issuers create and maintain an adhoc firewall to exclude "poor" people, effectively reaching the same result.

This is the answer, judge it based on accuracy because it isn't advice and it doesn't factor in anybody's cultural predilections, none of my answers do.