Here is a slightly different approach to zwon's where I only read the file in one place and use state variables to keep track of where we are in the data file.

use strict;
use warnings;

my $frgFile = q{spw768164.frg};
open my $frgFH, q{<}, $frgFile or die qq{open: < $frgFile: $!\n};

my $inFRG = 0;
my $inSeq = 0;

while( <$frgFH> )
{
    if( $inFRG )
    {
        if( m|^\}| )
        {
            $inFRG = 0;
        }
        elsif( $inSeq )
        {
            if( m{^\.} )
            {
                $inSeq = 0;
            }
            else
            {
                print;
            }
        }
        elsif( m{^acc:(\d+)} )
        {
            print qq{>$1\n};
        }
        elsif( m{^seq:} )
        {
            $inSeq = 1;
        }
        else
        {
            ;
        }
    }
    else
    {
        next unless m|^\{FRG|;
        $inFRG = 1;
    }
}

close $frgFH or die qq{close: < $frgFile: $!\n};
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The output.

>1101077781160
acaaggctggagtatttttttgtttagtaatttatttaattcagtttttatattttcataaactttttta
ggatcaccagggccattacttaaaaaaaaaccatcaaaatttctattaattatatcctcagcattaaaat
tgatctttagagagaaacttacctttgaaaatatattttttgttataaattaaatatccgttttgataat
taagtttagttttattatctaatacgggcatattaaatcatgtgtattagtatattatatcaaaggaaat
tcaaatgagtttggcaaaaaaatttctgatgacgttaaagtgctttaaaaggcggagatagaaaaacttt
catagcaaggtatgtctattctgagttaaaaattttctattaaagaaatctagagagagacgtgcttaat
tatctgacga
>1101077781161
gcgtgacgtttgagcagaagaattatttattaatttctgaggattttaagtctttaaaacaaaacgtttc
attcaaatttcaaaatcttgattataaagaagcgatggcactaatggctgagattggcaatattaatata
caactggctcagcagtattaaaaattgaactagatgcgttggaatctcaagggttaggaagaacagtttc
taatccaaattgtttaccctagacaatcaaactgccagtattaaac
[download]

I hope this is of interest.