There have been many threads on the different types of hosting, i.e., comparing free, shared, reseller, VPS and dedicated. Perhaps I can summarize:
Free: Low cost (no cost but host can advertise on your website); likely to be very limited as to services on offer; no support; service outages can be expected; host can disappear
Shared: Low cost; most services available; "unlimited" offers abound but the hidden limit (CPU usage, RAM, storage) can be "tight"; support depends upon host, service outages should be rare
Reseller: Same as shared but higher cost due to multiple control panels
VPS (Virtual Private Server): Higher cost as you have your own server (which is shared with other VPS users); limits on RAM, CPU, storage & bandwidth are the limiting factors; better support (IF managed VPS)
Dedicated: Most expensive as the entire server is dedicated to you use; best support (IF managed)
From this description, I think you are beyond the Shared/Reseller categories. Indeed, even the VPS with sufficient RAM to support the surges may be strained by the excessive demands of Joomla (it's been reported to be a memory hog - something I can't confirm or deny). That leaves dedicated which, by the funding that client is burning on his TV ads, is probably the most cost-effective way to go (and well within the client's budget). Just be SURE to get good support from your host as VPS/dedicated need a babysitter 7/24/365.
Just my 2¢ worth (well within budget)!