The usage of 'know' with object and object complement

No.

• *I know him a doctor.

is ungrammatical. Not all verbs take part in this construction, 2-place with B-Raising (see John Lawler's article). To illustrate:

[I consider him] [He is a friend]. The object of the first statement twins as the (suppressed) subject of the second statement: B-raising (... to subject). [I consider him a friend]. (The object of the second simple statement becomes an object complement in the process.)

Note that 'regard' doesn't have this property: an as is required.

As seen, one that does is consider

• I consider him a friend, a grammatical deletion of
• I consider him to be a friend.

But 'know' needs the 'to be', undeleted.

The basic structure you're asking about in your first example is a noun with a defining relative clause

• designs that / which they know work.

where the relativiser 'that' is often deleted, especially in conversation

• designs they know work.

This includes the use of previous designs that they know work ― both designs that they have created themselves and those that others have created.

Is this something like know A B (like "I know him(A) a doctor(B)" ?

No. Perhaps the easiest way to explain it (I don't deal in abstruse grammatical theory) is as follows:

The basic structure of the sentence is:

This includes the use of previous designs that work.

We can now insert the parenthetical "we know"

This includes the use of previous designs that (we know) work.

That last form is simply a contraction of

This includes the use of previous designs that, as we know, work.