On Localbitcoins I would only consider a sale of that size to someone with thousands of transactions and peerless feedback. And that would mean that they were a professional broker who more than likely would be looking to pay below the market rate. There should be more than a few who have that much money available.
On the plus side there would be no possibility of triggering some unknown limit as can happen with exchange and you're more than just some ticket number.
If I had a GDAX account I think I'd still lean towards using that providing I was fully verified on there and my limits were clear. You could also look into Gemini and their daily auction or an OTC sale via somewhere like Genesis Trading but their minimum sell might add up to more.
Interesting, thanks for a concise response. Just out of curiosity, even at the volume I am trading right now, would you recommend I use GDAX over Localbitcoins?