I'd go straight to Math::MPFR (which uses the mpfr library, which in turn uses the gmp library).
However, in doing so I'm plugging one of my own modules - which perhaps lessens the value of my advice.
Anyway, on my Windows box:

C:\_32>perl -MMath::MPFR=":mpfr" -le "Rmpfr_set_default_prec(1024001);
+$x = Math::MPFR->new(2); $x **= 1024000; print $x;" >out.txt
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The output file reveals:

5.194693363199925.....490149376e308254
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The result is instant (maybe a quarter of a second).
The length of the string that is output correlates nicely to yours. It is 308263, which is 308255 + 1 (for the decimal point) + 7 (for the "e308254").
Do we also agree on the value ?

Maximum precision for my mpfr library (and hence for my Math::MPFR) is 1073741823 bits.

Update: Just noticed that the OP now specifies that full precision is not required:

perl -MMath::MPFR=:mpfr -le "Rmpfr_set_default_prec(28);print Math::MP
+FR->new(2) ** 1024000;"
5.194693363e308254
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Cheers,
Rob