I wonder which type of if-conditional it belongs to. Grammar books tell me that there are three types of if-conditional.
That is a distorted picture presented by many course books and grammar books written for students. As Mario Rinvolucri wrote once (unfortunately I can't track down the article), English has over fifty types of conditionals. You'll find a more accurate picture of conditionals in a series of articles beginning
here.
This sentence with the word "should" resembles the second type but I guess the sentence does not belong to the second type as it is not about a "very unlikely" situation.
I suggest you stop worrying about what types of conditional sentences are, and concentrate on what degree of certainty they are suggesting, and what the time reference is. The form 'would like' (and its equivalents in some other Indo-European languages) may appear to suggest a condition, but it is, for practical purposes, very similar in meaning to the present simple form of WANT.
Moreover, grammar books tell me that there shouldn't be any modal verb in the if-clause, but this one has the word "would."
Many of the modals appear frequently in
if- clauses.