Have you tried the REST API? The GET sequence/id endpoint pulls out a sequence with just the ID. For example, http://rest.ensembl.org/sequence/id/ENSMUSP00000137272?content-type=text/x-fasta;type=protein.

pyGeno is also you friend. It does not require access to a REST API so it is more reliable and faster if you have a lot of proteins.

from pyGeno.Genome import *
ref = Genome(name = 'GRCh37.75')
prot = ref.get(Protein, id = 'ENSMUSP00000137272')[0]

And you also get all the information supplied by Ensembl for free: prot.gene.biotype, prot.transcript.sequence, prot.transcript.exons etc.

Actually, once you know which adaptor to use, it is quite simple. Here's a perl script that does it for an input text file with protein identifiers on each line:

#!/usr/bin/env perl

########################################################################### 
# script to download all the protein sequences from a list of identifiers

use strict;
use warnings;
use Bio::EnsEMBL::Registry;
use Bio::EnsEMBL::ApiVersion;
printf( "The API version used is %s\n", software_version() );

my $list=$ARGV[0];
print "Parsing IDs from $list\n";
open(LIST,"<$list")||die "Can't open $list\n";
my (@IDs);
while(<LIST>){
  chomp($_);
  push(@IDs,$_);
}

# Load the registry automatically
my $registry = 'Bio::EnsEMBL::Registry';
$registry->load_registry_from_db(
  -host=>'ensembldb.ensembl.org',
  -user=>'anonymous', 
);

open(PROT,">$list\_out.fa")||die "Can't open $list\_out.fa\n";

foreach my $ID (@IDs) {
  print PROT ">$ID\n";
  my $seqmember_adaptor = Bio::EnsEMBL::Registry->get_adaptor('Multi','compara','SeqMember');
  # fetch a Member
  my $seqmember = $seqmember_adaptor->fetch_by_stable_id($ID);
  print PROT $seqmember->sequence(),"\n";

}