be rid of / get rid of

I agree with your analysis.

It is possible for rid to be a pure adjective = clear/cleared of something that is unwanted.[1] It is now very rare as a true adjective. If it is accepted that “rid” is an adjective, then “of Karen” is an adverbial prepositional phrase modifying “rid” and “rid of Karen” is a complement, and there is no phrasal verb.

The alternative is that, as you say, “rid” is a participle modifier acting (with “of Karen”) as a predicate complement.

The OED also suggests that both to be rid of, and to get rid of are phrasal verbs with slightly different meanings:

P1 a. to be rid of: to be freed or relieved from (a troublesome or unwanted thing or person).

2000 M. Barrowcliffe Girlfriend 44 iii. 91 I consider everyone as a life partner—apart from Gerrard, of course. One day I will be rid of him.

P2. to get rid of: to remove or dispose of (a troublesome or unwanted thing or person).

2002 Woodworker Aug. 91/2 Smooth any rough edges to get rid of splinters.

I would disagree a little with this as to be indicates a state, whereas to get/ become indicates a change of state – as you say, both are copular.

[1] OED: Rid (adj.) rare. That has been ridded or cleared. Also rid-up. 1628 in H. Paton Dundonald Parish Rec. (1936) 246 He saw Robert Bowman fall on the rid land.

1866 C. Kingsley Hereward the Wake II. viii. 137 ‘We will make room for you! We will make a rid road from here to Winchester!’ shouted the Meeting, with one voice.

(You will see that rid expresses the idea of something that has been emptied of unwanted things.)