HoH will require the spouse to be an NRA.  Could be an idea if that fits the taxpayer's circumstances.

In my experience, a majority of green card applications for employment visa holders are sponsored by employers, even though there may be conditions for bonds or retention.  Whether they filed MFS, MFJ, or HoH in certain years have not been a material concern with these applications so long as its within the law, AFAIK.

Potential problems tend to arise after they receive their green cards when it comes to taking certain positions on their returns for §911 and treaty benefits.