They both have their problems and both are best avoided. The first attempts to join the two clauses I was a small kid and My classmates laughed at me at the time. They make good sense individually, so you might conclude that joining them with whose would also make good sense. Unfortunately, it doesn’t. Relative pronouns have antecedents, and the antecedent in (1) is undoubtedly a small kid. By the time we reach me in the sentence we’re set up to expect a reference to something other than the subject of the first clause, such as . . . laughed at everything I did.
The second attempts to join the two clauses I was a small kid and My classmates laughed at him at the time. They don’t make very good sense individually and joining them doesn’t improve matters. There’s again a mismatch between the first person subject of the main clause and the third person him in the subordinate clause. In certain contrived prose styles it might work, but for normal purposes it places too much of an interpretative burden on the reader.