Can had inversion occur in past tense, not past perfect?

Solution 1:

In short, yes, but such a usage will be considered stuffy.

In the prologue of Canterbury Tales, in the introduction of the Dr. of Phisick, we see

Full ready had he his 'Pothecaries, To send him druggis and 'lectuaries,...

Samuel Richardson's 1748 epistolary novel Clarissa, Or: the History of a young Lady, has this:

And thus, as Mr. Lovelace thought fit to take it, had he his answer from my sister.

Modern inversions of this sort are typically a way of expressing conditional mood.

We have this in a 1965 Harlan Ellison review of the movie The Train in Cinema magazine:

Unlike most of the flea-marketeers of Hollywood, director John Frankenheimer is a man who would deal with whales, had he his choice.

Piers Anthony's book Aliena Too has

Had he his choice of a perfect world, he would never have left her. But that kind of choice he never had.

You're unlikely to see this outside of poetry and literature, however. It's a little stuffy, verging on idiomatic, because in ordinary conversational English one would typically say "If I had my choice" rather than "Had I my choice".

Also see this old EL&U question.