I need to minimize [....] MySQL queries [....], or I end up using more than 25% of the server's CPU resources.  When that happens, they suspend my account and give me a nice e-mail/phone call.
Ouch!  Are you slammed by the 
hosting hammer for 
mere spikes, or can you rely on 
some patience being applied?  Odd, isn't it, that ads promoting shared-hosting don't ever seem to describe the provider's 
penalties for exceeding its configured limits on your sharing?  I know 
I hadn't thought about 
that before.
I'm on a shared hosting account, believe it or not.
That's actually 
easier to believe, because that suggests to me that you're being bitten by a weakness of the 
Apache model for shared hosting, where every site hosted runs as the same 'user', which likely deprives the operating system or server-subsystem of information needed to 
gracefully throttle the performance of sites, by lowering the execution-priorities of those users or tasks exceeding CPU-usage limits.  Or perhaps it's something completely different; I know far less about its & *nix internals than I'd like.
Or perhaps the hosting service-providers actually 
plan not to provide graceful throttling of performance, because providing it would eliminate a substantial selling point for upgrading to a dedicated hosting account.
I'd like to keep it [....] on a shared hosting account.
I assume that you're thinking mostly of the 
beeeg increase in monthly costs to indulge in 
dedicated hosting.  And elsewhere you 
have mentioned children, who presumably need feeding at least occasionally.