I have never used Net::OAuth but after a quick read through the docs, I think the correct answer is to put $content in extra_params and use to_post_body in the post method call.

my $ua = LWP::UserAgent->new;

my $content = { example => "content", name => "me" }

my $request = Net::OAuth->request("protected resource")->new(
       consumer_key => <key>,
       consumer_secret => <secret>,
       request_url => 'https://someplace.whatever',
       request_method => 'POST',
       signature_method => 'HMAC-SHA1',
       timestamp => time(),
       nonce => 'kllo9940pd9333jh',
       token => <token>,
       token_secret => <secret>,
       extra_params => { foo => 'bar', %{$content} }
);

$request->sign;

my $res = $ua->post(
       $request->to_url,
       Content_Type => 'form-data',
       Content => $request->to_post_body,
);
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I found a post from 2011 which seems to indicate that I am on the right track: https://txlab.wordpress.com/2011/09/03/using-tumblr-api-v2-from-perl/

Update: Another option would be to use Net::OAuth::Simple which seems basically do the same thing behind the scenes.