There's nothing I can do in my wrapper class to make: UNIVERSAL::isa($object_of_my_class, "IO::File") return true.

Nothing? Nothing reliable, perhaps, but try

*UNIVERSAL::old_isa = *UNIVERSAL::isa;
*UNIVERSAL::isa = sub {
    my($self,$pkg) = @_;
    return 1 if ref($self) eq 'MyClass' and $pkg eq 'IO::Handle';
    return UNIVERSAL::old_isa(@_);
};

package MyClass;
sub new { bless {} }

my $object_of_myclass = new MyClass();

my $rc = UNIVERSAL::isa($object_of_myclass, 'IO::Handle');
print $rc ? "It worked\n" : "Oh nuts\n";
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And no, I'm not serious.