Using "as" instead of "that" (I don't know as this is valid)
Your two examples become less liable to questions of interchangeability when they are both properly parsed as absolute constructions with ellipsis. When an absolute in sentence-initial position merely describes a state, such as being wealthy or a wealthy man, the participle is often omitted:
19.10.2017. The astute politician that he is, Mr Odinga has been using allegory and riddles to tell his supporters how the electoral commission denied him of victory. — “Raila Odinga - the man Kenyans either love or loathe,” BBC News.
In other contexts, the participle is usually expressed:
In a skewed way, Kathy's actions had forced Cory to make a better life for himself, but being the man he was, he would have done it anyway. — Gail Sattler, Dating the Best Man, 2015, 171.
and especially when the participle is not used absolutely:
Robbed of control by virtue of being a woman and the particular woman that she is, the anorexic, like the self-mutilating woman, has seized control over the one area that women are encouraged to control, and she is holding on to it desperately.— Bonnie Burstow, Radical Feminist Therapy: Working in the Context of Violence, 1992.
Your second example omits both the initial participle and the first conjunction in an adjectival as (adj.) as construction, which may or may not be expressed:
As intelligent as he is, he misjudges the evil of the others, believing they can be spoken with rationally and never realizes the seriousness of the situation and the potential for a deathly response. — Harold Bloom, William Golding's Lord of the Flies, Bloom’s Guides, 2004, 14.
Though still current after say, the non-standard, highly colloquial as introducing a relative clause in the Daniel Boone citation was far more frequent and occurred in more contexts in earlier centuries than today:
‘Why, I can’t say as I know any real harm of her,’ said her mother, ‘unless it’s as she’s a Catholic.’
‘Totty Nancarrow a Catholic!’ exclaimed Ackroyd. ’Why, I never knew that.’
‘Her mother was Irish, you see, an’ I don’t suppose as her father thought much about religion. I dessay there’s some good people Catholics, but I can’t say as I take much to them I know.’ — George Gissing, Thyrza, 1887.Dealing with the blues, but can’t say as I’m suffering. Joseph M. Monks, filmmaker, born Queens NYC, Blog post, 8 Feb. 2017.
Someone named se16teddy posted about this at forum.wordreference.com:
The OED has this usage as No. 26 of as, adverb and conjunction.
- Introducing a noun clause, after say, know, think, etc.; = that conjunction 1a.
It marks this usage: Now England regional (southern) and U.S. regional.
The OED's examples include the following:
1484 Caxton tr. G. de la Tour-Landry Bk. Knight of Tower (1971) lxiiii. 94, I saye not as ye shalle be pryuely and allone one by other.
1856 H. B. Stowe Dred xi. 100, I don't know as you'll like the appearance of our place.
1997 F. Chappell Farewell I'm bound to leave You 11, I don't know as I believe that talk.
It sounds quaint and folksy to my ear but not uncontemporary (is there a better word for that?).