How to do BLASTP for short sequences (<20 aa) effectively?
You should increase the -evalue 10 default option to a higher value because this value it's adjusted to the sequence database size. Check this table, maybe this can help you to decide which parameters are the best option for you.

    OPTION      TASK          DEFAUL    DESCRIPTION
    word_size   blastp        3         Word size of initial match.
    word_size   blastp-short  2         Word size of initial match.
    gapopen     blastp        1         Cost to open a gap.
    gapopen     blastp-short  9         Cost to open a gap.
    gapextend   blastp        1         Cost to extend a gap.
    gapextend   blastp-short  1         Cost to extend a gap.
    matrix      blastp        BLOSUM62  Scoring matrix name.
    matrix      blastp-short  PAM30     Scoring matrix name.
    threshold   blastp        11        Minimum score to add a word to the BLAST lookup table.
    threshold   blastp-short  16        Minimum score to add a word to the BLAST lookup table.
    window_size blastp        40        Multiple hits window size, use 0 to specify 1-hit algorithm.
    window_size blastp-short  15        Multiple hits window size, use 0 to specify 1-hit algorithm.