The short answer is you need a comma in each case.
I personally find the 'independent clauses' rule a bit unhelpful, although I know this is the standard way to explain when to use commas before 'but'. The reason being, one replaces a punctuation problem ("do I need a comma?") with a grammar problem ("are these independent clauses?") and one is no further forward. When dealing with sentences such as yours, which are somewhat abbreviated and fragmentary, such as might appear in the short descriptions of a menu, things can get hard to work out.
An alternative rule, to me, might be:
If "but" can be replaced with "notwithstanding", "on the other hand" or "on the contrary" precede it with a comma.
This means, if you are joining two "coordinate elements" that are linked but contrasting, use a comma.
All four of your sentences are saying the same thing, but with more or fewer implied words.
It is a Caesar salad, but [contrary to what one might expect] with fried noodles.
It is a Caesar salad. Notwithstanding, it contains fried noodles.
Nothing but Caesar salad will do.
No sooner had I started my Caesar salad but Jim walked in.
Cannot be rewritten as:
Nothing notwithstanding Caesar salad will do.
No sooner had I started my Caesar salad on the other hand Jim walked in.
Whether you explicitly include "it is" and "it contains" to make full sentences doesn't alter the need for a comma.
Most if not all other uses of 'but', which are usually considered not to require commas, appear to lack this explicit contrasting conjunction sense.
Anyway - this is my theory. Interested to hear if others think my 'rule' works.