EDIT: As mentioned by @dan and others, now I believe the correct answer is we need 是 in the sentence to change the meaning of 应该:
应该 should (requirement) → 应该是 should be ~ might be (possibility).
Discussion for the role of 是 in the sentence without 应该:
After consulting with the book Charles N. Li, Sandra A. Thompson, Mandarin Chinese: A Functional Reference Grammar, I think I have finally understood the purpose of 是 in the sentence:
I caught a cold.
In the Edit I mentioned there were two distinct interpretations: 是 was acting as copula and 是 was for indicating emphasis (but I was referring to some mistaken structures). Now I believe the actual solution is a combination between both.
According to the book, 是 is the copula verb, which can be employed in three types of constructions:
1. Simple copula sentences:
Subject + 是 + Noun phrase.
2. Special affirmative sentences:
Subject + 是 + Verb phrase or
是 + Sentence.
3. Presentative sentences:
Time phrase or locative phrase + 是 + noun phrase.
I believe the example could be either 1. or 2 depending on 感冒了 acting as a verb or as a noun. I will start with 2 because it feels the best interpretation.
2. Special affirmative sentences
The copula 是 (pronounced with stress) can be used to mean It is true that... or It is that... with respect to a statement already mentioned in the conversation.
For example, if we compare:
I don't have money.
It is true that I don't have money.
Without 是 the statement is neutral, we are just providing information or answering a question; but with 是 we are affirming what was previously suspected or inferred is true. 是 is still the copula verb linking the subject noun phrase to a full verb phrase. (As a side note: this use of 是 is not restricted only to link to verb phrases but it can also be placed before an entire sentence).
Probably this is the same purpose for our example if we understand 感冒了 as the verb caught a cold. The 是 is optional and it only provides an affirmation / emphasis for It is true that I caught a cold.
1. Simple copula sentences
These sentences typically have a subject that is concrete noun phrase that is linked by the copula verb 是 to a general noun phrase, which allows to identify or characterize the subject. For example:
She is my friend.
Here, the concrete subject 她 is linked to the general noun phrase 我的朋友 to indicate a trait.
However, there are some sentences called 'illogical' sentences, which can have a metaphorical meaning. For example:
Could mean I am Wang (literal), I vote for Wang (metaphorical), Wang is my favorite singer (metaphorical), I am a member of Wang's group (metaphorical), etc.
It does not mean I am a fried rice (literal), it indeed implies (e.g. in a restaurant) I am the one having the fried rice (metaphorical).
Similarly, if we understand 感冒了 catch-cold like a noun, 我是感冒了 does not mean I am a catch-cold (literal) but means I caught a cold (metaphorical).