I have a suggestion, as a former headhunter for 15 years:
Get your best résumé knocked out and on that person's desk TOMORROW. You can always say you were
updating your LinkedIn crap, but didn't want them to wait.
If you need to redo parts of your qualification so they read like you would have made the web page,
then do so, but don't fumble around. They want to see what you can do, and since every page has
an up time, it just so happens that you were in the midst of updating it - but your résumé is READY.
Email it and follow it up with a phone call - and ask when a good time to meet would be. Take charge
of the situation so that you'll know you have been quick to respond (a show of interest) and you'll
validate their interest in YOU by your quick response.