I think that I would splurge a little on memory and use a hash (or maybe an array if the names are really '1' .. '9'). All the operations you want are easy and straightforward with a hash:

my %state;
@state{1..9}=();
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1. have name, clear bit

   $state{$name} = undef;
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2. have name(s), set bit(s)

   @state{@names} = (1) x @names;
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3. get count of set bits

   my $count = grep {$_} values %state;
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4. have one set bit, get its name

   my $setname = (grep {$state{$_}} keys %state)[0];
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5. have multiple set bits, get names (debugging only)

   my @setnames = grep {$state{$_}} keys %state;
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6. bit-wise logical ops, OR, AND, ...

   # same as ordinary perl
   print $foo if $state{$one} && $state{$tother};
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A small hash like this isn't going to break the bank, and the perl operations to maintain it are much simpler than bit-twiddling.

Fixed typo in #2, thanks blokhead++