Sentence № 1 is punctuated correctly, and Trask’s advice is perfectly compatible with the advice of Harcourt Publishers.
The question poses a false dilemma. You have read the advice you found in Writer’s Harbrace Handbook as a blanket prohibition of a comma after a coordinating connector. But that’s not so. The advice is that a comma is not necessary or desirable for the purpose of separating a coordinating connector from the clause that follows.
I read “it is incorrect to use a comma after a coordinating conjunction” to mean that when you write such a conjunction you should not on that account then write a comma. If a comma is necessary in that place for a reason having nothing to do with the conjunction, then the advice is irrelevant and should not be followed slavishly.
In the example, there isn’t a comma after the connector for that reason: your eyes deceive you. The clause that follows the connector in the example is “cats scare me witless”. That is, before inserting the parenthetic expression, the connected sentence has no comma there:
I like dogs, but cats scare me witless.
After inserting the parenthetic expression, the meaning is likely to be misconstrued without using bracketing commas to set it off as recommended by Dr Trask. These commas are not there on account of the conjunction, so they do not offend Writer’s Harbrace Handbook:
I like dogs, but<, I am embarrassed to admit in the presence of
polite company,> cats scare me witless.