Is there any way to turn a string of arbitrary length text into an arbitrary BigInt and back?

Not that I know of - though I don't usually use Math::BigInt.
Is it guaranteed that the "arbitrary length text" will contain only valid base 64 characters ?

Math::GMPz (plug) will certainly handle such text, even if it contains invalid base 64 characters - though Math::GMPz isn't included in perl, and depends upon the gmp C library:

use strict;
use warnings;
use Math::GMPz qw(:mpz);

my ($order, $size, $endian, $nails) = (1, 1, 0, 0);
my $s = 'Som3/Arbitr@ri!y/long/string';
my $z = Math::GMPz->new();

Rmpz_import($z, length($s), $order, $size, $endian, $nails, $s);

print $z, "\n";
$z++;
print "$z\n";

my $incremented_s = Rmpz_export( $order, $size, $endian, $nails, $z);
print "$s\n";
print "$incremented_s\n";

# Next retrieve original string:

$z--;
my $orig_s = Rmpz_export( $order, $size, $endian, $nails, $z);
print "$orig_s\n";

__END__
Outputs:
8786758437493627904416651010636024986357833879283932790893327248999
8786758437493627904416651010636024986357833879283932790893327249000
Som3/Arbitr@ri!y/long/string
Som3/Arbitr@ri!y/long/strinh
Som3/Arbitr@ri!y/long/string
[download]

But perhaps someone else can provide a solution that better suits your needs.

Cheers,
Rob.